NEW INSTRUMENTS AND METHODS OF MEASUREMENT

MONOCHROMATORS

A. S. Toporets

INTRODUCTION

The radiation of light sources usually consists of a more or less broad spectrum of frequencies, and the task of monochromatization is to select from the given radiation spectrum some interval of finite width.

The concept of “monochromatic light” requires clarification, since the content that experimental practice puts into this concept is considerably broader than what follows from the theoretical definition.

Monochromatic light, in the strict theoretical sense, is associated with oscillations of a single frequency and of infinite duration. In reality, as experience shows, all known radiations possess neither the one nor the other attribute. The reasons lie in the properties of the very substance emitting the light.

From the point of view of quantum theory, for strictly monochromatic radiation to arise it is necessary that the energy levels between which the transition accompanied by radiation takes place be ideally narrow; however, even in the case of an isolated atom not subject to external influences, the energy levels have a width different from zero, and the radiation of the atom over a certain interval of time will be blurred within certain limits \(\Delta \lambda\).

A line is usually assigned a natural half-width, which is the spectral interval (situated to the right and to the left of the maximum) at whose boundaries the intensity of the line is equal to half the intensity at the maximum. Classical optics gives for this quantity the expression

\[ \Delta \lambda = 1.2 \cdot 10^{-4}\ \text{Å}. \]

This quantity could also be regarded as limiting for the width of the spectral interval. However, there are still other causes that affect the width of a spectral line and are determined by the conditions of existence of the emitting substance. First, the line contour is blurred owing to the Doppler effect; second, this is also promoted by the interaction of the emitting atoms, which leads to a certain uncertainty in the position of the atom’s energy levels, i.e., to a broadening of the line.

Thus, radiation cannot be monochromatic in the strict sense of the word by its very nature. Experimentally observed lines have a half-width of the order of \(10^{-2}\,\text{\AA}—10^{-1}\,\text{\AA}\). These values should be regarded as the lower limit of the possible monochromatization of light.

Since monochromatic light is understood to mean light waves with frequencies lying in a frequency interval of finite width, it becomes necessary to introduce a measure of the monochromaticity of light that would make it possible to characterize quantitatively the spectral purity of monochromatic beams, and at the same time the monochromatizing ability of the instrument by means of which they are obtained.

No established measure of monochromaticity exists. The concept of resolving power, adopted for the characterization of spectrographs, is not applicable to monochromators. The width of a spectral interval, expressed in frequencies or in wavelengths, likewise cannot characterize the degree of monochromaticity of light, since, being identical in absolute magnitude, this width will have different relative significance in different regions of the spectrum. It is therefore expedient to use the concept of relative spectral width, which for an infinitely narrow spectral interval is expressed as the ratio

\[ \frac{d\lambda}{\lambda}. \]

For a spectral interval of finite width one may write

\[ \mu = \int_{\lambda_1}^{\lambda_2} \frac{d\lambda}{\lambda} = \ln \frac{\lambda_2}{\lambda_1}. \tag{1} \]

In the ideal case, when \(\lambda_1 = \lambda_2\), \(\mu = 0\). Hence it is clear that \(\mu\) represents the deviation from ideal monochromaticity of the light. The smaller this quantity, the closer the monochromaticity of the given beam is to the ideal.

If the spectral interval is measured in frequencies, then the quantity \(\mu\) introduced by us is expressed as

\[ \mu = \ln \frac{\nu_1}{\nu_2}. \tag{2} \]

From a comparison of (1) and (2) it is clear that, on the frequency scale as well, \(\mu\) has the same expression as on the wavelength scale.

Thus, as a measure of the monochromaticity of light we shall take the logarithm of the ratio of two wavelengths (or frequencies) conventionally bounding the given spectral interval.

The production of monochromatic beams is carried out by various methods. All of them are based on the interaction of light and matter and can be encompassed by the following well-known phenomena: a) dispersion, b) diffraction, c) interference, d) selective absorption, e) selective reflection. All these methods are used for the monochromatization of light; however, their effectiveness and prevalence are far from identical. Thus, diffraction spectral instruments, for a number of reasons, have not yet found wide use, although they possess high resolving power. The phenomenon of interference is currently used in so-called interference light filters, which make it possible to isolate comparatively narrow spectral regions. Glass and gelatin filters with selective transmission (absorption) find wide application in those cases where very narrow spectral intervals are not required, for example, in absorptiometric installations.

Table I

	\(\mu\)
1. Instruments of high resolving power . . .	up to \(10^{-6}\)
2. Prism monochromators . . .	\(10^{-4}—10^{-3}\)
3. Focal monochromators . . .	\(10^{-2}—10^{-1}\)
4. Interference filters . . .	\(10^{-2}—10^{-1}\)
5. Filters with selective transmission	\(10^{-1}\)

Finally, considerably more advanced and widely used are instruments based on the dispersion of light by matter. These are the so-called monochromators.

In Table I the various methods of obtaining monochromatic light are compared according to their monochromatizing capacity, which is expressed through the quantity \(\mu\) introduced earlier.

It is evident from the table that monochromators occupy second place. These instruments make it possible to solve a whole series of important problems, among which the following may be mentioned:

a) absolute measurements of the spectral distribution of the radiation energy of light sources;

b) measurement of the spectral sensitivity of receivers of radiant energy;

c) determination of the coefficients of transmission, reflection, and scattering;

d) excitation of luminescence;
e) photochemical investigations;
f) photoelectric investigations.

There is no doubt that in the future of spectroscopy, in connection with the successful development of photoelectric methods for recording weak light fluxes, monochromators will play no less important a role than spectrographs.

In connection with the rapid development of spectrophotometric methods of analysis in the most diverse fields of science and technology, monochromators are becoming ever more widespread as an integral component of any spectrophotometer. Meanwhile, there is no special description of monochromators. The present review aims to fill this gap and is an attempt to systematize material published mainly in the journal literature.

I. SIMPLE MONOCHROMATORS

The purpose of a monochromator is to transform a beam of white light into a discontinuous or continuous sequence of spatially separated monochromatic rays and to select from this sequence a certain spectral interval. The spatial separation of monochromatic beams and the possibility of a continuous transition from interval to interval essentially distinguish a monochromator from various kinds of filters. In the latter, the set of frequencies, limited by the transmission band of the filter, is the same for all points of the beam cross-section; in other words, at every point of this cross-section one can encounter all the frequencies of the given limited spectral sequence, whereas in a beam emerging from a monochromator the different frequencies are spatially separated. Some overlap of neighboring frequencies does occur in this case as well, owing to the fact that one always has to deal with beams of finite width and with imperfect optical systems; however, with sufficiently narrow slits and a corrected optical system, the superposition of some frequencies on others can be made negligible.

To obtain a pure spectrum it is necessary that the angles of incidence, as well as the angles of emergence from the prism, of the entire set of rays forming the given monochromatic beam be identical. For a prism with plane faces this condition can be fulfilled if the rays incident on the prism face are parallel. Therefore, an integral part of a spectral instrument—in our case, a monochromator—is a system of lenses or mirrors, the projection system, whose purpose is to ensure the correct path of rays in the instrument.

The simplest optical layout of a monochromator is as follows: entrance slit, lens (or mirror), prism, lens (mirror), exit slit (Fig. 12). The entrance slit limits the incoming beam; the first lens makes the beam parallel; the prism disperses the white beam into monochromatic components; the second lens gathers these beams in the plane of the exit slit, forming a spectrum, from which the exit slit cuts out a narrow spectral interval.

Thus, in a monochromator one may distinguish two systems: the dispersing system (prism) and the projection system (lenses, mirrors). In most spectral instruments these two systems are separate; however, instruments are also known in which both functions—dispersion and projection of the beams—are performed by a single system (cf. III).

Projection system

The projection system may be dioptric or catoptric. In the first case lenses are used, simple or compound; in the second, mirrors. There is a whole series of monochromator models in which simple lenses are employed, but, as is well known, they have a number of shortcomings—for example, spherical and chromatic aberrations. All these shortcomings cause a broadening of the spectral interval, i.e., they impair the degree of monochromatization of the light. Therefore, in the best models, compound objectives are used, consisting of two or even three components, corrected with respect to one or two of the indicated shortcomings. It should be noted that even the use of corrected systems is not a radical solution to the problem, since in practice correction can be achieved only for a comparatively limited portion of the spectrum. Therefore the so-called achromats and even apochromats justify their name only in a comparatively narrow spectral region[^1]. Outside this region they, like simple lenses, require additional focusing.

Recently, lenses with aspheric surfaces, free from spherical aberration, have come into use. The advantages of such lenses over corrected compound ones are that a smaller amount of material is required and light losses in the instrument are reduced, since the thickness of the lenses and the number of reflecting surfaces are decreased.

Of course, such lenses are not free from chromatic aberration and likewise require special focusing when the instrument is set for each wavelength.

A catoptric system differs advantageously from a dioptric one in that it has no chromatic aberration, owing to which one and the same projection system can be used over a more

over a wider range of wavelengths than in the case of a dioptric system. Thus, in the infrared region mirrors have gained a monopoly and are hardly likely ever to lose it. At the same time it must be pointed out that in instruments with catoptric systems one always has to deal with oblique beams, and this is associated with the appearance of astigmatism and coma. Another drawback of these systems is an insufficient reflection coefficient.

In the visible region of the spectrum, where this coefficient may reach values of 0.9 and even higher, mirrors are not inferior to lenses with respect to light losses, but in the ultraviolet region lenses have a certain advantage in this respect. The available indications in the literature concerning the production of layers with a high reflection coefficient² make it possible to hope for the production of catoptric systems that are little inferior to dioptric systems also in the ultraviolet region. Experience in constructing instruments with a catoptric system shows that the use of these systems can be extended at least to 200 mμ¹⁹.

From this brief comparison of dioptric and catoptric systems one may conclude that future monochromators must be constructed either with aspherical optics and automatic focusing, or with mirror optics. In this connection two problems arise: 1) establishing the production of retouched lenses and mirrors, and 2) finding methods and coatings for obtaining mirrors with a high reflection coefficient.

Dispersing System

The principal part of a monochromator—the prism—performs an important function: it decomposes the white beam incident upon it into monochromatic beams.

The angular dispersion of a prism is expressed by the formula

\[ \frac{dD}{d\lambda} = \frac{2\sin \frac{A}{2}} {\sqrt{1-n^{2}\sin^{2}\frac{A}{2}}} \frac{dn}{d\lambda}. \tag{3} \]

Here \(D\) is the angle of deviation of the ray, \(A\) is the refracting angle of the prism, and \(n\) is the refractive index of the substance from which the prism is made³. From expression (3) it is evident that, in order to obtain a large angular dispersion, one must use a prism with a large refracting angle and a material with a large dispersion \(\left(\frac{dn}{d\lambda}\right)\).

Thus, the materials from which prisms are made must be transparent in the spectral region of interest and must possess large dispersion. These two requirements are difficult to satisfy simultaneously, owing to the fact that, as a rule, the dispersion curve has a large steepness in that part of the spectrum which

begins to be appreciably absorbed and, conversely, in the region of greatest transparency the dispersion is insignificant. Usually, for different regions of the spectrum, prisms of different materials are used. Thus, for the visible and near infrared regions various kinds of glass are used; in the ultraviolet and infrared—quartz (crystalline and fused), fluorite, and crystals of alkali-halide salts (LiF, NaF, NaCl, KCl, KBr). The latter are now being grown artificially[^4][^5]. Figures 1 and 2 give the dispersion curves of these substances. From these curves it is evident that some crystals of alkali-halide salts have, in the ultraviolet region, a dispersion several times greater than that of quartz and fluorite. The named crystals have, compared with quartz, the further advantage that they are isotropic. However, there is one essential circumstance which at present limits the wide use of crystals of alkali-halide salts: most of them are hygroscopic; the only exceptions are lithium fluoride and sodium fluoride. In this connection the question of protection arises. Attempts to cement thin quartz plates onto the working faces of the prism by optical contact have, for a number of reasons, not solved the problem of protection. The solution must apparently be sought in finding thin transparent films that are impermeable to moisture and sufficiently strong. The favorable optical properties of these crystals, which greatly expand the possibilities of instrumen—

Fig. 1. Dispersion curves of various substances in the ultraviolet and visible regions of the spectrum.

...structure—transparency over a broad spectral range, high dispersion, and a variety of refractive indices—should serve as an incentive for the earliest possible solution of the protection problem.

The most common are prisms with a refracting angle of \(60^\circ\). Prisms with a larger refracting angle are not used, for the reason that this is associated with an increase in reflection losses and with a decrease in the beam cross section, i.e., a decrease in luminosity.

In addition to ordinary triangular prisms with refracting angles of \(60^\circ\) and \(30^\circ\), monochromators use a constant-deviation prism and a Féry prism. The constant-deviation prism consists of two 30-degree prisms cemented to the legs of a right-angle prism (Fig. 3). At minimum deviation it changes the direction of the emerging ray, relative to the incident one, by \(90^\circ\), independently of the refractive index; as a result it received its name. A ray entering such a prism, after passing through the first refracting prism at minimum deviation, is totally reflected from the rear face of the prism and passes through the second prism likewise at minimum deviation. The angles of incidence and emergence of the ray from the prism are equal, whence it follows that the angle of deviation of the ray is \(90^\circ\).

Fig. 2. Dispersion curves of certain substances in the infrared region of the spectrum.

Fig. 3. Constant-deviation prism.

Glass prisms of this type are often made from a single piece with angles of \(60^\circ\), \(90^\circ\), \(75^\circ\), and \(135^\circ\).

In making prisms from quartz it is necessary to take into account the presence of double refraction in quartz. To eliminate the harmful effect of double refraction, the prism is cut from a quartz crystal in such a way that the optical

axis lay in the principal section of the prism parallel to its base. Then rays passing through the prism at the angle of minimum deviation do not undergo double refraction. Quartz also exhibits rotation of the plane of polarization. To avoid the splitting of lines associated with this phenomenon, the prism is made of two equal halves of right-rotating and left-rotating quartz (Fig. 4). Prisms of constant deviation are made of quartz comparatively rarely.

The Fery prism[^6] is a combination of a concave mirror with a prism, in which both working faces are curved (Fig. 5).

Let \(R\) be the radius of curvature of the entrance surface of the prism \(MPQN\). Place the slit at the point \(C\), chosen so that rays emerging from it and striking the points \(P\) and \(Q\) of the prism surface form equal angles \(i\) with \(R\).

Fig. 4. Prism made of quartz.

Fig. 5. Fery prism.

After refraction, rays of different wavelengths strike the second, mirror surface at different angles, lying within the range of the prism’s total angular dispersion. For some wavelength this angle will be such that the rays fall on the rear surface of the prism normally. It is easy to see that, on emerging from the prism, these rays will all converge at the point \(C\), as at a focus. Rays of other wavelengths will have foci at points adjacent to \(C\). If the rays refracted at the points \(P\) and \(Q\) are prolonged to their intersection, then the point \(B\) will be the center of curvature of the rear surface of the prism. It can be shown that all the points mentioned, \(P\), \(Q\), \(A\), \(B\), and \(C\), lie on one circle, whose radius is equal to \(\dfrac{R}{2}\), and which is the focal line.

A feature of the Fery prism is that it performs both functions—dispersion and projection of the beams—and therefore does not require a projection system. However, it has astigmatism, i.e., it stretches the point image of the slit into a vertical line.

Spectral Interval

The basic requirements that must be imposed on a monochromator are as follows:

a) a high degree of monochromatization of the light,
b) a high power of the luminous flux emerging from the instrument,
c) constancy of the geometry of the beam passing through the instrument.

The first two requirements are mutually exclusive: increasing the degree of monochromatization is inevitably associated with a decrease in the power of the monochromatic luminous flux, and conversely. Therefore, when choosing the design of the instrument, one must take its purpose into account.

Thus, from a monochromator intended for measurement purposes, what is required above all is a sufficient degree of monochromatization; as for the power of the luminous flux, with present-day possibilities for registering weak luminous fluxes this requirement no longer plays the role that it did only a few years ago. Now, for most tasks, there is no longer any acute necessity to build instruments with high luminosity because of insufficient sensitivity of energy receivers.

On the contrary, one may expect that monochromators intended for the investigation of narrow spectral lines will be built with low luminosity, in order better to satisfy the first of the stated requirements.

Instruments intended for studying very weak radiations or for obtaining powerful monochromatic beams, such as are required, for example, for photochemical and luminescent purposes, cannot isolate narrow spectral bands. Only in combination with powerful sources of line radiation with sparsely arranged lines can such monochromators satisfy both of the stated requirements.

The third requirement, constancy of the geometry of the beams, is especially important for instruments intended for absolute measurements of energy. Therefore, in these instruments special attention must be paid both to the perfection of the optics and to the quality of the mechanical system that determines the displacement of the individual elements of the instrument (rotation of the prism, displacement of the slit, focusing of the lenses).

Let us consider on which structural elements the width of the spectral interval isolated by the instrument depends, for a given slit width.

The linear dispersion of an instrument having focal length \(f\) of the second collimator can be found from the formula:

\[ \frac{dl}{d\lambda} = \frac{dD}{d\lambda}\cdot f = \frac{2\sin\frac{A}{2}} {\sqrt{1-n^3\sin^2\frac{A}{2}}} \frac{dn}{d\lambda}\,f . \tag{4} \]

Hence the spectral interval

\[ \Delta\lambda= \frac{\sqrt{\,1-n^{2}\sin^{2}\frac{A}{2}\,}}{2\sin\frac{A}{2}}\, \frac{d\lambda}{dn}\,\frac{1}{f}\,\Delta l . \tag{5} \]

If the entrance slit has width \(a\), then, for a magnification equal to unity, radiation with wavelengths lying in the interval will fall at each point of the spectrum lying in the plane of the second slit:

\[ \Delta\lambda_{1}= \frac{\sqrt{\,1-n^{2}\sin^{2}\frac{A}{2}\,}}{2\sin\frac{A}{2}}\, \frac{d\lambda}{dn}\,\frac{a}{f}. \tag{5a} \]

But since the exit slit has width \(a'\), the monochromatic beam emerging from the instrument will have a spectral width

\[ \Delta\lambda= \frac{\sqrt{\,1-n^{2}\sin^{2}\frac{A}{2}\,}}{2\sin\frac{A}{2}}\, \frac{d\lambda}{dn}\,\frac{1}{f}\,(a+a'). \tag{6} \]

Such is the width of the spectral interval for slits of finite width in the case of ideal optics. But owing to the fact that even in instruments with corrected projection systems aberrations occur, the image of each point of the entrance slit will in reality be blurred. If the spherical aberration gives a circle of confusion of diameter \(b\), and the curvature of the lines due to astigmatism and distortion is characterized, at a given height of the slit, by a sagitta \(c\), then the actual width of the spectral interval will be determined by the formula

\[ \Delta\lambda= \frac{\sqrt{\,1-n^{2}\sin^{2}\frac{A}{2}\,}}{2\sin\frac{A}{2}}\, \frac{d\lambda}{dn}\,\frac{1}{f}\,(a+a'+b+c). \tag{7} \]

The influence of aberrations on the width of the spectral interval is illustrated by Fig. 6. The rectangular contour gives the energy distribution and the width of the spectral interval calculated theoretically from formula (6); the curve gives the same quantities, determined experimentally by means of a linear thermopile for a slit width of \(0.3\) mm. From the figure it may be concluded that, for the instrument studied, the aberrational spot has a diameter of about \(0.3\) mm.

Thus, the width of the spectral interval is proportional to the width of the slits and to the magnitude of the aberrations, and inversely propor-

... to the material’s dispersion \(\left(\dfrac{dn}{d\lambda}\right)\) and to the focal length of the objective of the second collimator. By using prisms with large dispersion, as well as long-focus objectives, it is possible to obtain sufficiently narrow spectral intervals. The same result can be achieved by reducing the slits; however, along this path, in the case of a line spectrum there is a limit below which it is inexpedient to go. The width of the entrance slit should not be less than the diameter of the aberration spot*). It is easy to see that, for \(a \ll b+c\), the width of the spectral interval practically does not decrease with reduction of the entrance slit, while the brightness of the monochromatic beam falls sharply. This can be seen from Fig. 7, which shows the change in brightness of a monochromatic line \((\lambda = 546\,m\mu)\) when the slit width is varied. When the slit is increased from \(0.05\) mm, the brightness grows and at \(0.3\text{--}0.4\) mm reaches a maximum. A further increase of the slit beyond the value \(a' = a+b+c\) does not change the brightness, but only changes the magnitude of the flux. This essential fact should always be kept in mind when using a monochromator with a line source of light. In the example considered, the slit should not be made smaller than \(0.3\) mm.

Fig. 6. Influence of aberrations on the width of the spectral interval.

Fig. 7. Change in beam brightness with change of slit width.

The situation is somewhat different when using sources of continuous spectrum. In this case, even with a slit width smaller than the diameter of the aberration spot, uniform brightness can be obtained at the exit slit. Indeed, a spectral interval of width \(\Delta\lambda\) is spread out at the exit slit over a width \(a+b\); in this case, at the edges of the slit there will be an overlap of blurred neighboring portions of the spectrum. As a result, the brightness over the whole width of the exit slit will not differ from the theoretical value, but the purity of the spectrum will suffer. The “contamination” of the spectrum by aberrations will be determined by the quantity \((b+c)\dfrac{d\lambda}{dl}\). Even with very narrow slits, when \(a \ll b+c\), the width of the spectral

*) With ideal optics the minimum slit width is determined by the Rayleigh criterion\(^3\).

Table II

(Focal distance — 500 mm, slit width — 0.1 mm)

\(\lambda\) (mµ)	Quartz: \(\dfrac{dn}{d\lambda}\) (mµ\(^{-1}\))	Quartz: \(\Delta\lambda\) (mµ)	NaCl: \(\dfrac{dn}{d\lambda}\) (mµ\(^{-1}\))	NaCl: \(\Delta\lambda\) (mµ)	TF-3: \(\dfrac{dn}{d\lambda}\) (mµ\(^{-1}\))	TF-3: \(\Delta\lambda\) (mµ)	TF-5: \(\dfrac{dn}{d\lambda}\) (mµ\(^{-1}\))	TF-5: \(\Delta\lambda\) (mµ)
200	\(125\cdot10^{-5}\)	0,18	\(500\cdot10^{-5}\)	0,035
250	62	0,38	140	0,16
300	33	0,74	70	0,34
350	19	1,3	35	0,69
400	\(125\cdot10^{-6}\)	2,0	\(230\cdot10^{-6}\)	1,1	\(520\cdot10^{-6}\)	0,38	\(550\cdot10^{-6}\)	0,3
500	62	4,0	120	2,1	220	0,90	260	0,72
600	37	6,8	70	4,0	120	1,7	150	1,2
700	27	9,7	37	6,8	80	2,6	80	2,4
800	20	12,8	25	10,2	50	4,2	54	3,6
1000	14	18,0	15	17,0	25	8,4	32	6,6

of the interval cannot be made smaller than a certain value, namely:

\[ \Delta\lambda_{\min}= \frac{\sqrt{\,1-n^{2}\sin^{3}\frac{A}{2}\,}} {2\sin\frac{A}{2}}\, \frac{d\lambda}{dn}\,\frac{1}{f}\,(b+c). \tag{8} \]

Table II gives the dispersion values of various substances and the calculated widths of the spectral interval for monochromators with prisms \((60^\circ)\) made of these substances. It is clear from the table that for NaCl the dispersion in the ultraviolet region is, on average, twice as large as for quartz. Consequently, rock salt, with respect to dispersion, is considerably more advantageous than quartz, and not only in the ultraviolet but also in the visible region of the spectrum. The same must also be said of glasses (TF-3 and TF-5), for which the dispersion in the visible and near infrared regions of the spectrum exceeds the dispersion of quartz by three to four times\(^{33}\). It follows that the use of quartz prisms in instruments designed for a broad spectral region (for example, from \(0.2\) to \(2\mu\)) is inadvisable: the degree of monochromatization at the edges of the spectral region differs by two orders of magnitude.

Monochromatic flux and transmittance of the monochromator

The transmittance coefficient is a very important characteristic of a monochromator, since, for a given power of the light source, the magnitude of the monochromatic flux emerging from the instrument and having a definite spectral width \(\Delta\lambda\) will depend on the optical properties of the structural elements of the instrument. The attenuation of the light flux in the monochromator will be determined by reflection and scattering at the surfaces of the optical system, absorption within the individual elements (lenses and prisms), and vignetting.

The transmittance coefficient of a monochromatic flux of extremely narrow spectral width is defined as the ratio of the flux emerging from the instrument to the flux entering the instrument, i.e.,

\[ T_\lambda=\frac{\Delta\Phi'_\lambda}{\Delta\Phi_\lambda}. \tag{9} \]

At the same time, for a monochromator one may introduce the concept of throughput, which will characterize what part of the radiation of the light source in a given spectral interval the monochromator transmits. This quantity is determined by the geometry of the beam and by the transmittance coefficient, namely:

\[ P_\lambda=G\cdot T_\lambda, \tag{10} \]

where \(G\) is the geometrical factor. In general form the magnitude of the geometrical factor is expressed by the relation

\[ G=n^2\frac{S_1s_1}{f_1^2}. \tag{11} \]

Here \(n\) is the refractive index*), \(S_1\) is the utilized aperture area of the objective of the first collimator, \(f_1\) is the focal length of this objective, and \(s_1\) is the area of the entrance slit. In other words, the geometrical factor is the product of the aperture ratio of the instrument and the area of the entrance slit.

The magnitude of the monochromatic flux passing into the instrument may be expressed as

\[ \Delta\Phi_\lambda = B_\lambda \frac{S_1s_1}{f_1^2}. \tag{12} \]

The same flux at the output

\[ \Delta\Phi'_\lambda = B_\lambda \frac{S_1s_1}{f_1^2}T_\lambda . \tag{13} \]

Bearing in mind that \(G\) is an invariant, one may write:

\[ \Delta\Phi'_\lambda = B_\lambda \frac{S_2s_2}{f_2^2}T_\lambda . \tag{13a} \]

Formula (13) is valid only for strictly monochromatic light. But since at the exit slit there is a superposition of neighboring regions of the spectrum, through each point of the exit slit there passes a luminous flux with wavelengths contained in the interval \(\Delta\lambda_1\), determined by formula (5a). Therefore the brightness of the outgoing beam will be proportional to \(\Delta\lambda_1\):

\[ B_{\Delta\lambda_1}=\int_{\lambda_1}^{\lambda_2} B_\lambda\,d\lambda=\overline{B}_\lambda\cdot\Delta\lambda_1 . \]

Here \(\overline{B}_\lambda\) is the average brightness within \(\Delta\lambda_1\). The flux will be equal to:

\[ \Delta\Phi'_\lambda=\overline{B}_\lambda \frac{S_1s_1}{f_1^2}T_\lambda\Delta\lambda_1 . \tag{13b} \]

The experimental determination of the transmission coefficient can be carried out both for natural light and for light polarized parallel and perpendicular to the refracting edge of the prism.

Usually a monochromatic beam obtained with the aid of another monochromator is passed into the monochromator under investigation. In the absence of vignetting, the entire flux, except for that part which will be lost in the instrument owing to unavoidable reflections at the surfaces and absorption inside the optical

*) Since most instruments operate in air, \(n\) will henceforth be omitted.

details will pass into the second collimator of the instrument and fall on the exit slit. What, then, must be the minimum dimensions of the slit so that this entire flux can leave the instrument? In the case of ideal optics and equality of the focal lengths of the collimator objectives, the exit slit should have the same dimensions as the entrance slit. In a real instrument, however, the exit slit should have width

\[ a' = a + b + c \]

and height

\[ h' = h + \Delta h, \]

where \(b\) and \(c\) are the quantities defined earlier, and \(\Delta h\) is the magnitude of the linear stretching of the point image in the meridional plane due to the astigmatism of the system*).

Figure 8 gives the results of measurements of the transmission coefficient of the ISP-17A monochromator with mirror optics (see the schematic in Fig. 13)**). From these data it can be seen that the losses reach 47–65%. For the individual elements making up the instrument, these losses are distributed as follows:

reflection at the prism faces . . . . . . 18–25%
absorption in the prism . . . . . . . . . 0–15%
losses upon reflection from mirrors . . . 29–25%

Fig. 8. Transmission coefficient of the monochromator.

As can be seen, a substantial fraction of the losses consists of losses associated with the dispersing prism (up to 40%). Losses due to the projection system are also considerable. It is true that in monochromators with a dioptric system (single-lens objectives) they are somewhat lower (18–20%). Usually in monochromators uncoated optics are used; meanwhile, coating could noticeably reduce reflection losses both from the prism and from the lenses.

The calculation of the transmission coefficient in general form is quite complicated, since the individual rays in the beam pass through differ-

*) It is appropriate to note here that when working with a line-emission source and rare lines, one can obtain larger fluxes \(\Delta \Phi_\lambda\) with monochromators of medium or even small aperture by increasing the area of the entrance slit to dimensions at which the lines almost touch.

**) The measurements were carried out by Kh. L. Pes’kina.

paths in absorbing media. Below is an analysis of this question, carried out by Perry^8.

The beam of light in a monochromator is stopped down either by the prism (Fig. 9, a), or by the collimator objective (Fig. 9, b), or by both the prism and the objective (Fig. 9, c). The transmittance in the latter case may be regarded as the mean of the two limiting cases (a) and (b).

Let the \(x\)-axis intersect the axis of the collimator and lie in the plane of the principal section of the prism; \(r\) is the radius of the objective and \(\theta\) is the angle between the radius and the \(x\)-axis.

Fig. 9. Stopping down of the beam by the prism and the objective.

Let \(\dfrac{d\Phi_\lambda}{d\sigma}\) denote the flux density for the interval \(\Delta\lambda\) in the wavelength region \(\lambda\), per unit transverse section of the beam \((\sigma)\); let \(F\) denote the transmittance calculated by Fresnel’s formulas for the whole system; let \(\tau\) denote the transmittance of the prism material per unit length; and let \(l\) denote the path of the ray in the prism. It can be shown that \(l\) will be a linear function of \(x\), namely

\[ l=ax+b=ar\cos\theta+b. \]

The constants \(a\) and \(b\) are readily determined; the zero of \(\theta\) is chosen so that \(a\) is positive. If \(x_+\) and \(x_-\) are the limiting values of \(x\), and \(\pm Y\) are the limiting values of \(y\) for case \((a)\), then the flux incident on the prism for cases \((a)\) and \((b)\) is expressed as follows:

\[ \Delta\Phi_a=2Y(x_+-x_-)\frac{d\Phi_\lambda}{d\sigma}, \]

\[ \Delta\Phi_b=\pi r^2\frac{d\Phi_\lambda}{d\sigma}. \]

The flux finally passing through the instrument will be

\[ \Delta\Phi'_a = 2Y\frac{d\Phi_\lambda}{d\sigma}F\tau^b \int_{x_-}^{x_+}\tau^{ax}\,dx, \]

\[ \Delta\Phi'_b = 2r^2\frac{d\Phi_\lambda}{d\sigma}F\tau^b \int_{0}^{\pi}\tau^{ar\cos\theta}\sin^2\theta\,d\theta. \]

After introducing the new variable

\[ z=\frac{1}{2}a(x_+-x_-)\ln\tau=ar\ln\tau \]

and integration, the following expressions are obtained:

\[ T_a=\frac{\operatorname{sh} z}{z}\tau^b F, \tag{14} \]

\[ T_b=\frac{2J_1(z)}{z}\tau^b F, \tag{15} \]

where \(J_1\) denotes the Bessel function of the first kind and first order.

For the intermediate case \((в)\) the transmittance coefficient is obtained as the average of these two,

\[ T_c=\left(\frac{\operatorname{sh} z}{2z}+\frac{J_1(z)}{z}\right)\tau^b F. \tag{16} \]

The expression in parentheses is always greater than unity, and since the constant \(b\) is the mean path length (at \(x=0\)), the transmittance for the full aperture is always greater than the transmittance for the mean path length \(b\). With very small absorption of light by the prism material, the transmittance coefficient \(T\) becomes equal to \(F\). This latter quantity can be calculated for the angle of least deviation by the formula

\[ F=\frac{1}{2}\left[ \prod_{i=1}^{k}\left\{\frac{\sin^2 2i}{\sin^2(i+i')}\right\}_{i} + \prod_{i=1}^{k}\left\{\frac{\sin 2i}{\sin^2(i+i')\cos^2(i-i')}\right\}_{i} \right]\times \]

\[ \times \left[ \prod_{i=1}^{j}\left\{\frac{4n}{(n-1)^2}\right\}_{i} \right] \prod_{i=1}^{l} R_i . \tag{17} \]

Here \(i\) and \(i'\) are the angles of incidence and refraction of the rays in the prism, \(R\) is the reflection coefficient for the mirrors included in the system. The products extend over all surfaces: \(k\) prisms, \(j\) lenses, and \(l\) mirrors.

In formula (17) it is assumed that the angles of incidence and refraction of the rays in the lenses are small, and therefore the formula for normal incidence may be applied. This is the case only in instruments with low relative aperture. In instruments with high relative aperture \((1:2)\), the angles of incidence of peripheral rays may reach \(70^\circ\), which leads to an attenuation of the transmitted beam by 20–30%. Thus, the quantity \(\frac{d\Phi}{d\sigma}\) in the space behind the first objective of the collimator is a certain function of \(r\), decreasing toward the periphery of the beam, which is easily observed in practice in instruments with high relative aperture. This circumstance must be taken into account in the theoretical comparison of dioptric and catoptric systems with respect to the transmittance coefficient, since in such a comparison perpendicular incidence of light on the lens surface is assumed, and as a result underestimated values of the losses are obtained.

Designs of Monochromators

Monochromators may be distinguished according to several features: by spectral region, by projection system, and by design.

The range of application of the most widespread monochromators covers the wavelength interval from 0.2 to 15 μ. The distinction between monochromators used in the ultraviolet, visible, and infrared regions is, in essence, reduced to the difference in the materials from which the dispersing prisms and lenses are made, if the projection system is dioptric. In this case the choice of material is determined by two conditions: transparency and dispersion. In short, within the indicated wavelength interval the spectral region of application of monochromators is not a fundamental criterion for classifying monochromators. The same may also be said of the projection system, since in principle the use of dioptric and catoptric systems is possible within the indicated range. Only the presence of chromaticism in lenses and the difficulties of focusing them force one to prefer a catoptric system to a dioptric one in the infrared region.

A substantial distinguishing feature of monochromators is their design. One may distinguish instruments: a) autocollimating with an ordinary prism, b) autocollimating with a Féry prism, c) with a constant-deviation prism, and d) with the Wadsworth arrangement.

Fig. 10. Diagram of an autocollimating monochromator.

The autocollimating system is distinguished by its simplicity (Fig. 10), since in it a semiprism replaces a whole prism and one objective (a lens or mirror) performs the role of both objectives. Setting to a wavelength is carried out by rotating the prism; at the same time, each time the condition of minimum deviation is satisfied.

In instruments of low and medium aperture this system gives good results and makes it possible to control the slits by means of one and the same mechanism.^9 When quartz is used, autocollimation is advantageous also because it relieves the designer of the combination of right- and left-rotating quartz necessary in making a 60-degree prism. A substantial drawback of the autocollimating system is the relatively large amount of scattered light, which is inevitable because the dispersed beam travels in the same space as the incident beam.

Instruments with a Féry prism^10 are also built on the principle of autocollimation; however, they differ from the ordinary arrangement of this type by still greater simplification: the prism performs both func-

tion—dispersion and projection of beams. In this sense such monochromators have a similarity to focal monochromators. The kinematic scheme of an autocollimation system with a Féry prism can be implemented according to a scheme resembling, in its principle, the Eagle mounting for a concave diffraction grating (Fig. 11). The change in the angle of incidence for rays of different wavelengths is achieved by moving the prism with a special mechanism that provides both a minimum of deviation and focusing of the beam at the exit slit.

Instruments with a constant-deviation prism (Fig. 12) are used in the visible and, more rarely, in the ultraviolet region of the spectrum[^11]. The projection system here is usually dioptric, sometimes achromatized, since a rational combination of a catoptric system with a constant-deviation prism is very

Fig. 11. Diagram of a monochromator with a Féry prism.

Fig. 12. Diagram of a monochromator with a constant-deviation prism.

difficult. Monochromators of this type are widely used because of the convenience of adjustment and the simplicity of handling them.

The Wadsworth scheme[^12] has found wide application in monochromators for the infrared part of the spectrum. It is somewhat reminiscent of the scheme with a constant-deviation prism. In both cases reflection of the ray occurs: in the first case—from a mirror, in the second—from the rear face of the prism. The only difference is that in the Wadsworth scheme the reflection occurs after or before the prism, while in the constant-deviation prism it occurs between the half-prisms. In addition, there is some difference in the loss of light upon reflection: the constant-deviation prism is more advantageous.

Fig. 13. Wadsworth scheme.

The path of the ray and the arrangement of the elements of the Wadsworth scheme are shown in Fig. 13. The three-sided prism is rigidly connected with a plane mirror and is located on a rotating table. The ray passing through the prism at minimum deviation falls on the mirror and, being reflected from

of it, travels in some direction which makes with the direction of the incident ray an angle \(\beta\), related to the angle \(\psi\) by the following relation:

\[ \beta = 180^\circ - 2\psi. \]

It is clear from the figure that \(\psi\) is the angle between the plane bisecting the refracting angle of the prism and the plane of the mirror. Since the prism and the mirror are rigidly connected, when the whole system is rotated the angle \(\psi\) remains constant. Hence it follows that the angle \(\beta\) also remains constant for any values of the angle of minimum deviation. It follows from the formula that, when the angle \(\psi\) changes from \(0\) to \(90^\circ\), the angle \(\beta\) changes from \(180\) to \(0^\circ\). For values exceeding \(90^\circ\), the angle \(\beta\) becomes negative, and a position is possible in which the mirror becomes parallel to the ray, i.e. the ray will travel in the direction in which the prism deflects it. Obviously, in this case the Wadsworth arrangement is inapplicable.

Wadsworth showed\(^{13}\) that a ray passing through the arrangement at the angle of minimum deviation undergoes no lateral displacement when the table is rotated only in the case where the axis of rotation of the table is made to coincide with the line of intersection of the plane of the mirror and the plane bisecting the refracting angle of the prism. M. A. Yur’ev gave a simple proof of this proposition and showed what displacement of the ray may occur when the line of intersection does not coincide with the axis\(^{14}\).

Fig. 14. Arrangement of mirrors in a monochromator: \(a\) — crossed, \(b\) — zigzag.

In monochromators with a catoptric system, the arrangement of the mirrors is of great importance. This question was investigated qualitatively by Czerny and co-workers\(^{15,16}\). It turned out that coma, which appears in the case of oblique beams, can be eliminated if the mirrors are arranged in the appropriate manner. Figure 14 shows two possible arrangements of mirrors in a monochromator. We shall call them, in what follows: \(a\) — crossed, and \(b\) — zigzag (\(z\)). Let us first consider the crossed arrangement.

Let \(F\) be a point source of light. From it there propagates a spherical wave, which the mirror \(S_1\) must transform into a plane wave. But since the light source is off the axis, the wavefront after reflection from the mirror will not be plane, but will be

have the form of a certain curved surface, whose trace is shown in Fig. 14. Indeed, if \(FO_1\) is equal to the principal focal distance of the mirror, while \(FA_1 \ll FO_1 < FB_1\), then the spherical wave will be transformed into a plane wave only in the part directly adjacent to \(O_1\). The part of the wave reflected from the mirror in the region \(A_1O_1\) will correspond to a diverging beam, and in the part \(B_1O_1\) to a converging one, i.e. in the first case the wave front will lag behind, in the second it will run ahead relative to the plane part. At the second mirror this effect will be intensified; as a result, the image of the point source will not be point-like, but blurred, and moreover to one side. It follows from this that, in order to obtain a point image from a beam incident on a spherical mirror at an angle to its principal axis, one must use not a plane wave, but a wave having the form shown in Fig. 14. This is precisely what occurs in the zigzag arrangement (Fig. 14, б). The first mirror distorts the wave front precisely in the way necessary for the second mirror to concentrate the beam into a point.

Fig. 15. Wadsworth scheme with cruciform arrangement.

Fig. 16. Pfund scheme.

From the standpoint of what has been said, the autocollimation system is not free from coma, since it is essentially a cruciform scheme. As for the Wadsworth scheme, two variants are possible in it. In the first (Fig. 15) coma is present, while in the second (Fig. 13) it is absent.

All that has been said applies only to one type of aberration—coma. As for spherical aberration and astigmatism, they arise independently of the method of arranging the mirrors. Dymke \(^{17}\) showed that astigmatism occurring in catoptric systems with oblique beams grows rapidly as the angle between the incident and reflected beams increases, and also as the relative aperture increases. In this case the blurring of the image occurs chiefly along the height of the slit; in width the image is blurred only slightly. Consequently, astigmatism of the projection system has a noticeable effect only on brightness and to a small degree on the purity of the spectrum.

Catoptric systems with a small amount of astigmatism are possible only at relatively large focal distances,

since only in this case can the angle between the incident and reflected beams be made sufficiently small (a few degrees). There also exists a scheme^18 in which, by introducing plane mirrors, this angle can be reduced to zero (Fig. 16). It is not difficult to see that this method of eliminating astigmatism is associated with a decrease in the transmission coefficient of the instrument.

II. DOUBLE MONOCHROMATORS

When working with ordinary monochromators one always has to deal with stray light, which appears in the instrument as a result of multiple reflection from optical components, scattering on dusty surfaces and inside prisms and lenses. This stray light is always admixed with the light that has passed through regularly, as a result of which the monochromatic beam emitted by the instrument is more or less “contaminated.” The amount of this parasitic light may be small, but when selective energy receivers are used, comparatively small “contaminations” of the spectrum can lead to large measurement errors. These errors will be especially large when working in that region of the spectrum where the sensitivity of the receiver to the regularly transmitted light is small.

To eliminate stray light, suitable filters are sometimes used. But this method is not very effective, since the pass band of filters is usually many times wider than the spectral interval isolated by the monochromator and, moreover, along with the stray light the regular light is also appreciably weakened. The most radical method of eliminating stray light is the use of double monochromators.

A double monochromator is a combination of two simple monochromators coupled in such a way that the monochromatic beam selected by the first part of the instrument passes once more through the same system in its second part.

Fig. 17. Double monochromator: dispersions of the same sign.

There are several models of double monochromators. They can all be divided into two basic types, differing in the character of their dispersion, more precisely, in the relative arrangement of the prisms in the two halves of the instrument.

The arrangement of the prisms may be asymmetric or symmetric. In the first case (Fig. 17) the beam of light entering the monochromator undergoes double dispersion; in the second, dispersion of the light occurs only in the first half, while the second acts as a filter excluding scattered light (Fig. 18).

In monochromators of the first type, the transition from one region of the spectrum to another is accomplished by simultaneous rotation of both prisms. It is easy to see that, for the complete exit of the beam selected by the instrument from the exit slit, the path of the rays in both halves of the instrument must be entirely identical. This is possible only if the prisms are perfectly identical and are rotated by a single mechanism.

In monochromators with a symmetric arrangement of prisms, light of the same spectral

Fig. 18. Double monochromator: dispersions are opposite.

interval is obtained at the exit slit as emerges from the middle slit. The transition from one region of the spectrum to another may be carried out in two ways. First, by moving the middle slit in the plane of the spectrum. The part of the spectrum cut out by the middle slit will fall on the exit slit. If the middle slit is removed altogether, white light can be obtained at the exit slit. Thus, an instrument of this type can also serve as a variator, making it possible to obtain mixed light of any spectral composition.

For this purpose it is sufficient, in the plane of the spectrum, to cover the unwanted spectral regions with rectangular screens.

The second method of moving along the spectrum, as in an instrument of the first type, is rotation of the prisms.

The greater complexity of a double monochromator in comparison with a single one imposes higher requirements both on the optical components and on the mechanical design. Indeed, the presence of a larger number of reflecting and refracting surfaces and the longer path of the ray make a change in the geometrical extent of the ray under mechanical displacements more probable. Therefore, when choosing the scheme

of a double monochromator one should prefer the one that permits a smaller number of mechanical displacements.

In existing models of double monochromators, all the optical arrangements described earlier are used: the autocollimation arrangement, the constant-deviation arrangement, and the Wadsworth arrangement. In addition, in a double monochromator the aforementioned variolluminator arrangement (Fig. 19) can be implemented, which differs substantially from those just listed and is not applicable in ordinary monochromators[^19].

Fig. 19. Optical scheme of a variolluminator (Toporets).

The advantage of this arrangement in comparison with others is that the transition from one region of the spectrum to another is accomplished

Fig. 20. Double monochromator with constant-deviation prisms.

only by a linear displacement of the middle slit, while all the components determining the path of the beam remain stationary. Thanks to this, the geometrical length of the rays inside the instrument remains constant. However, the conditions for the passage of different rays are not the same, and at the minimum of deviation only some single ray passes. As a result, the extreme portions of the spectrum are subject to vignetting, the more so the farther they are from the middle of the spectrum. Consequently, the spectral distribution of energy in the flux emerging from the instrument and entering it will differ strongly.

In Figs. 20–23 are shown the optical schemes of typical double monochromators used in laboratory practice[^20],[^21],[^22].

Fig. 21. Double monochromator with two Wadsworth arrangements.

Fig. 22. Double autocollimation monochromator.

The Spectral Interval Selected by a Double Monochromator

Let us determine the width of the spectral interval selected by a double monochromator. This question was analyzed in greatest detail by Van Cittert \(^{23}\) and Teren \(^{7}\).

Fig. 23. Double autocollimation monochromator.

We shall consider a double monochromator having a magnification equal to unity and free of aberrations.

The slits, more precisely their relative dimensions, play an essential role in the operation of a double monochromator.

Let us denote the entrance slit by \(F\), the middle slit by \(F'\), and the exit slit by \(F''\); their widths by \(a\), \(a'\), and \(a''\), respectively. Let the reciprocal linear dispersion at the middle slit be

\[ \frac{d\lambda}{dx}=2K. \]

In a monochromator of the first type, at the exit slit this quantity will be equal to

\[ \frac{d\lambda}{dx}=K, \]

since after passage through the second prism the same spectral interval will occupy, in the plane of the exit slit, a length twice as large as at \(F'\).

Fig. 24. Scheme of conjugation of slits:
\(a\)—in a monochromator of the first type,
\(b\)—in a monochromator of the second type.

We shall make use of a certain geometrical scheme \(^{7}\). Let us depict the slits by parallel line segments arranged so that the middle one is at equal distances from the outer ones. The lengths of the segments correspond to the widths of the slits (Fig. 24, \(a\)).

If the midpoints of the slits are conjugate for a definite wavelength, then the spectral interval passing through the exit slit \(F''\) will be bounded by the rays \(\lambda_1\) and \(\lambda_2\), emerging from the midpoint of the entrance slit \(F\) and passing through the extreme points of the slit \(F''\). The width of this spectral interval is determined as follows:

\[ \lambda_1-\lambda_2=Ka''. \tag{18} \]

But \(\lambda_1-\lambda_2\) is not an expression for the full spectral interval of the rays emerging from the instrument. In view of the fact that the slits \(F\) and \(F'\) have finite dimensions, in the plane of the slit \(F''\) there will be a superposition of neighboring wavelengths on one another, and through each point of the slit \(F''\) there will pass a spectral interval \(\Delta \lambda_1\), whose magnitude is determined by the width of the entrance slit \(F\) and by the linear dispersion, i.e.

\[ \Delta \lambda_1=2Ka \tag{19} \]

for the middle slit \(F'\), and

\[ \Delta \lambda_1=Ka \tag{20} \]

for the exit slit \(F'\).

Thus, the full spectral interval that has passed through the exit slit of a double monochromator of the first type will be equal to

\[ \lambda_1-\lambda_2+\Delta \lambda_1=K(a+a''). \tag{21} \]

This condition determines the sum of the widths of the two outer slits for a given spectral interval and linear dispersion. It is assumed that the middle slit does not diaphragm the beam. Its optimum width will be clarified subsequently.

In monochromators of the second type, each point of the slit \(F\) gives a spectrum in the plane \(F'\). Then, if nothing obstructs the light and the transmission coefficient of the instrument is equal to unity, all monochromatic rays passing through the slit \(F'\) will be collected at the slit \(F''\) in the same spectral composition (Fig. 24, b).

The width of the spectral interval \(\Delta \lambda_1\) in this case is also determined by the width of the slit \(a\); however, the multiplier must be the quantity \(2K\), since no subsequent decomposition takes place in the second half of the monochromator:

\[ \Delta \lambda_1=2Ka. \]

At the same time the difference of the mean wavelengths passing through the extreme points of the slit \(F'\) will be expressed as

\[ \lambda_1-\lambda_2=2Ka'. \tag{22} \]

Assuming that all the light which has passed through \(F'\) will also pass through the slit \(F''\), for the full spectral interval we obtain:

\[ \lambda_1-\lambda_2+\Delta \lambda_1=2K(a+a'). \tag{23} \]

Thus, the width of the spectral interval for monochromators of the second type is determined by the first half of the instrument. If, for example, slit \(F\) is very wide and slit \(F'\) very narrow, then in the plane \(F''\) a pure spectrum is obtained, whose length is equal to \(a\). Conversely, if \(F\) is narrow and \(F'\) is absent, then at \(F''\) an image of \(F\) is obtained, formed by rays of all wavelengths that have passed through the instrument. Thus, in monochromators of the type considered, the middle slit plays the role of a kind of filter.

Comparing (21) and (23), we see that a monochromator of the first type, with identical slits, is capable of giving a spectral interval twice as narrow as a monochromator of the second type.

Brightness at the exit slit and monochromatic flux

The question of the distribution of brightness at the exit slit is most conveniently considered using the same geometrical scheme as in the preceding paragraph.

It is quite obvious that this distribution will depend on the ratio of the widths of the three slits—\(a\), \(a'\), and \(a''\). The various combinations are exhausted by the four principal cases shown in Fig. 25. If the distance of the point under consideration at the exit of the beam from the middle of the exit slit is denoted by \(x\), then

Fig. 25. Brightness distribution at the exit slit.

the dependence of \(\Delta\lambda\) on \(x\) can be written for these cases as follows:

1st case: \(2a' > a + a''\); \(\Delta\lambda = Ka\).

2nd case: \(a > 2a' + a''\); \(\Delta\lambda = K2a'\).

3rd case: \(a < 2a' < a + a''\)

a) \(\Delta\lambda = Ka\) for \(0 < 2x < 2a' - a\);

b)
\[ \Delta\lambda=\frac{K(2a'+a-2x)}{2} \]
for \(2a' - a < 2x < 2a' + a\);

c) \(\Delta\lambda = 0\) for \(2a' + a < 2x\).

4th case: \(2a' < a < 2a + a''\)

a) \(\Delta\lambda = K2a'\) for \(0 < 2x < a - 2a'\);

b)
\[ \Delta\lambda=\frac{K(2a'+a-2x)}{2} \]
for \(a - 2a' < 2x < a + 2a'\);

c) \(\Delta\lambda = 0\) for \(a + 2a' < 2x\).

In the first two cases \(\Delta\lambda\), like the brightness, is constant over the entire width of the exit slit \(a''\); in the other two there exists a central region of uniform brightness, on both sides of which there are zones of gradually decreasing brightness. The exit slit may transmit all or part of the uniform zone (cases 1 and 2), or, in addition, wholly or partially weakened zones (cases 3 and 4).

Let us determine which of the cases considered is the most favorable for photometric measurements. Usually the energy receiver has some definite width, and for reliable measurements it is necessary that its entire receiving area lie in the zone of uniform brightness. It is clear from Fig. 25 that in the third and fourth cases one will have to be limited only to the zone \((2a' - a)\), smaller than \(a''\), and this is equivalent to the first and second cases. Thus, we are left to choose between the two latter cases. Let the width of the receiver be \(a''\), and let the sensitivity be such that for normal measurements a certain minimum energy is required, determined by the width of the spectral interval \(\Delta\lambda_1\).

In the first case

\[ a=\frac{\Delta\lambda_1}{K};\qquad 2a'=a+a''. \tag{24} \]

From the condition \(2a' > a + a''\) we have taken the minimum value of \(a'\), since a wider slit \(F'\) would increase the stray light (Fig. 26, a). The total length of the spectrum in the plane \(F''\) is equal to

\[ 2a' + a = 2\frac{\Delta\lambda_1}{K} + a'', \]

and the difference between the mean wavelengths \(\lambda_1\) and \(\lambda_2\) reaching the edges of the receiver will be

\[ \lambda_1-\lambda_2=Ka''. \]

In the second case \(\Delta\lambda_1\) is determined by the middle slit

\[ a'=\frac{\Delta\lambda_1}{K} \tag{25} \]

and the minimum width for \(a\)

\[ a=2a'+a''. \]

The length of the spectrum in the plane \(F''\), as in the first case, is

\[ 2a'+a=2\frac{\Delta\lambda_1}{K}+a'', \]

but the difference between the mean wavelengths reaching the edges of the receiver will be (Fig. 26, \(b\)):

\[ \lambda_1-\lambda_2=2Ka'', \]

i.e. it is twice as large as in the first case. On this basis the first case should be preferred.

Thus, if \(\Delta\lambda_1\), \(\dfrac{d\lambda}{dx}\), and the width of the exit slit \(a''\) are given, then the widths \(a\) and \(a'\) can be found from the relation

\[ a=\frac{\Delta\lambda_1}{K};\qquad a'=\frac{a+a''}{2}. \tag{26} \]

All that has been said applied to a double monochromator of the first type. In monochromators of the second type the situation is somewhat

Fig. 26. Dispersion and spectral interval.
\(a\)—in the first case; \(b\)—in the second case and \(v\)—in a monochromator of the second type.

different. Since the dispersion of light occurs only in the first half of the instrument, while the second half acts in the reverse direction, in the ideal case light of the same spectral composition and with the same brightness distribution as at the entrance slit will be concentrated at the exit slit. In other words, every point of the slit \(F''\) is always conjugate to some point of the slit \(F\) for any wavelength (Fig. 26, \(v\)). It follows that, with proper use of the instrument, it must be

\[ a=a''. \]

Through each point \(F''\) passes a spectral interval determined only by the middle slit (see Fig. 24):

\[ \Delta\lambda_1=2Ka', \]

and the difference of the mean wavelengths passing through the edges of the exit slit is

\[ \lambda_1-\lambda_2=2Ka''. \]

The luminous flux can be found in the following way. We saw above (136) that the brightness at the exit slit is proportional to \(\Delta\lambda_i\); in this case the flux will be proportional to \(\Delta\lambda_i a''\). For a monochromator of the first type we have:

\[ \Delta\Phi_{\Delta\lambda}=C\Delta\lambda_i a''=CK'a\cdot a''. \tag{27} \]

The maximum of this quantity is obtained for \(a=a''\), and by (26)

\[ a=a'=a'', \tag{28} \]

i.e. the luminous flux emerging from a double monochromator has its maximum value when all three slits are identical.

This condition is also optimal for a monochromator of the second type, for which the flux is expressed as

\[ \Delta\Phi'_{\Delta\lambda}=C\Delta\lambda_1 a''=C2Ka'a'' \]

or, for \(a=a'\),

\[ \Delta\Phi'_{\Delta\lambda}=C2Kaa''. \tag{29} \]

Comparing equations (27) and (29), one may conclude that, under identical conditions, a monochromator of the second type gives at the output a luminous flux twice as intense as that of a monochromator of the first type. It should be remembered, however, that the width of the spectral interval (compare (21) and (23)) obtained from a monochromator of the first type is half that obtained from a monochromator of the second type. Consequently, the first instrument will give the same degree of monochromatization with an entrance slit twice as wide (\(2a\)). It is easy to see that in this case the luminous flux will increase fourfold, i.e. it will be twice as great as in a monochromator of the second type. Thus, with respect to the two principal characteristics—the degree of monochromatization and the luminous flux—the double monochromator of the first type is more advantageous.

Causes that impair the brightness and purity of the spectrum

We saw above that the principal causes of losses are reflection and absorption occurring as the beam passes through the optical system of the monochromator. In a double monochromator the number of elements is twice as large, and consequently the light losses should increase approximately twofold. If both halves are completely identical, then the transmission coefficients of the double monochromator and of each of its halves are related by

\[ T=T_1^2. \]

a) Vignetting. However, a double monochromator is not simply a connection of two ordinary monochromators. Especially distinctive in this respect is the Van Cittert model²⁴, shown in Fig. 18. In this instrument the condition of minimum deviation is satisfied for only one wavelength; all the other rays do not pass along the axis of the instrument. As a result, vignetting occurs, especially for the most strongly deviated (extreme) rays. Usually, in order to eliminate this defect, which greatly weakens the transmission of the instrument, a collector is used, placed near the middle slit; but it is not possible to avoid vignetting completely in this way, since the deviation of the rays by the lenses does not fully compensate the deviations produced by the prism.

Fig. 27. Transmission coefficient of the Van Cittert double monochromator.

How large the vignetting is in the Van Cittert double monochromator can be seen from Fig. 27, which shows the values of the vignetting coefficient obtained by Terhune⁷ for rays of different wavelengths and at different distances in height from the middle of the slit. As is seen from the figure, the vignetting coefficient varies within considerable limits and is equal to unity only in a comparatively narrow spectral region and for a slit whose height does not exceed 2 mm.

b) Aberrations in a double monochromator are doubled in comparison with an ordinary one. Terhune and Devin²⁰ calculated the magnitudes of the aberrations for a double monochromator with a catoptric system, having a relative aperture of 1:9 and \(f = 55.6\) cm (Fig. 21). Table III gives the values of the spectral interval caused by various aberrations*).

*) The values given in the table are calculated for a slit having a height of 20 mm.

It is seen from the table that the greatest “contamination” of the spectrum is due to astigmatism. The elimination of astigmatism is possible only either by reducing the angle between the incident and reflected beams to zero (in the present instrument this angle was

Table III

$\lambda$ (m$\mu$)	Spherical aberration $\delta\lambda$ (m$\mu$)	Astigmatism $\delta\lambda$ (m$\mu$)	Curvature of the image $\delta\lambda$ (m$\mu$)	Coma*) $\delta\lambda$ (m$\mu$)
2000	1.8	16.2	3	—
1000	0.54	4.0	0.63	—
800	0.30	2.68	0.25	0.79
546	0.10	0.89	0	0.07
400	0.03	0.24	0.04	0.07
360	0.01	0.08	0.02	—

equal to $5^\circ$), or by the use of aspherical mirrors with a displaced axis. The manufacture of mirrors of this kind involves great difficulties; therefore, for the time being, they are not used.

Curvature of the lines is caused by the prism. With a finite height of the entrance slit its image is curved because the rays passing through the slit at some distance from the axis of the instrument intersect the prism at some angle to its principal section. Therefore, in order to admit the entire flux of a given spectral interval into the second half of the instrument, it is necessary to use a curved intermediate slit. If the slit is straight, then the spectral composition of the beam passing into the second half of the instrument will be different for different points of the slit along its height. A double monochromator of the first type gives, for a straight slit $F$, curved lines at $F'$ and lines twice as curved at $F''$. In order not to impair the purity of the spectrum, the entrance and exit slits are usually made curved with opposite curvature, and the intermediate slit straight. Of course, this measure is not absolutely effective for the whole region of the spectrum covered by the monochromator, since the curvature of the lines also depends on the wavelength, but the use of curved slits noticeably weakens the “contamination” of the spectrum. In the monochromator described by the French authors, the “contamination” of the spectrum from curvature of the lines is the same as that obtained from diffraction broadening of the image.

In monochromators of the second type the action of the prisms is mutually opposite; therefore the curvature produced in the first half of the instrument must be compensated in its second half. Consequently, in order to use the entire flux with straight

*) For the varioluminator scheme.

at the end slits, it is necessary to make the middle slit curved.

The elimination of coma in double monochromators with a catoptric projection system is achieved by the same measures discussed above. The use of a zigzag arrangement for each half is also preferable here to a cross-shaped one. It should be noted, however, that in double monochromators the cross-shaped scheme may also be used—provided that, in the arrangement of the components and in the path of the beams, one half is the mirror image of the other. In this case the coma produced in the first half will be eliminated in the second, but with equal slit widths contamination of the spectrum at the middle slit will occur.

III. FOCAL MONOCHROMATORS

An ordinary lens may be proposed as a simple monochromator. The idea of a device of this kind is clear from Fig. 28. Light passing through the first aperture and passing through

Fig. 28. Simplest scheme of a focal monochromator.

the lens will be gathered not at a single point, but on a certain segment of a straight line coinciding with the axis. This will occur because the short-wavelength rays, being refracted more strongly, will give an image closer to the lens, while the long-wavelength rays will give one farther from it. If, in this case, the paraxial rays are blocked by a screen, then, by moving a screen with an aperture along the axis, different portions of the spectrum can be emitted from the aperture.

The idea of focal monochromatization has long been known. Wood and Rubens first used a quartz lens for isolating long infrared waves[^25]. Subsequently this method was repeatedly used by various authors. Thus, G. A. Tikhov, with the aid of this method, photographed stars in monochromatic rays[^26]. A. N. Terenin[^27] used this method to isolate spectral lines shorter than 240 mμ, employing a quartz lens. For the same purposes and in the same region, American authors also used a quartz lens[^28]. Duncan applied the method of focal monochromatization in the Schumann region (174.5–156.1 mμ), using a fluorite lens[^29].

Despite its simplicity, the method of focal monochromatization has not found wide application, and until very recently there existed no instruments based on this principle. The explanation of this fact should be sought in that the most widespread substances used for the manufacture of dispersing systems do not possess sufficient dispersion.

The author of the present article proposed a design for a focal monochromator for the ultraviolet region of the spectrum,^30 in which the dispersing lens is made of sodium chloride, and the correcting one of lithium fluoride. The need to correct the system with respect to spherical aberration follows from the desire to obtain a high degree of monochromatization.

Let us consider the special features of the focal monochromator. A focal monochromator differs essentially from an ordinary prism monochromator. Like any spectral instrument, a prism monochromator performs two functions: 1) dispersion and 2) collection (projection) of light beams. The first function is performed by a prism with a constant refracting angle; the projection of the image is provided by objectives. In a prism instrument one strives, on the one hand, to increase the dispersion of the prism and, on the other, to eliminate the chromatism of the projection system. In a focal monochromator both named functions are performed by one and the same lens, as a result of which the “defect” of the lens—chromatic aberration—turns into a positive quality, and the need for achromatization disappears. In calculating the objective for a focal monochromator one has to solve a problem of an unusual character: correction of the objective in the sense of spherical aberration must not entail a decrease in its chromatism.

It turns out that this problem is not only feasible, but that in correcting the system for spherical aberration it is even possible to increase its chromatism somewhat.

Suppose there is a chromatic objective corrected with respect to spherical aberration. For simplicity of calculation let us consider an equivalent lens.

The lens may be regarded as a prism with a variable refracting angle, continuously changing from zero (on the principal optical axis) to some maximum value (at the edge of the lens). The magnitude of the refracting angle can be easily found. Namely, it is equal to the sum of the angles between the principal optical axis of the system and the radii of curvature drawn at the points of entrance and exit of the ray.

\[ A=\varphi+\varphi' \tag{30} \]

or, for equal curvature of the lens surfaces,

\[ A=2\varphi . \tag{30a} \]

In the case of a single lens, it is expedient to place the entrance aperture at the double focal distance for some wavelength (Fig. 28), since in this case the system entrance aperture—lens—exit aperture is symmetrical.

For the ray emerging from the aperture, in this case the condition of minimum deviation is satisfied; moreover, the angle of deviation of the ray is related to the aperture angles by the following relation:

\[ D = u + u', \tag{31} \]

or, in our case,

\[ D = 2u. \tag{31a} \]

The arrangement described presents considerable inconveniences in the operation of the instrument, since in passing from one region of the spectrum to another it is necessary to move the apertures relative to the lens, and this entails movement also of the illuminator and the objective, which must be situated at definite distances from the apertures.

It is more rational to make the system of two lenses, in such a way that the entrance aperture is in the principal focus of one lens, and the exit aperture in the principal focus of the other (see Fig. 31). Then adjustment of the instrument to a definite spectral region is achieved by simultaneous and symmetrical displacement of the lenses relative to the transverse axis of the system, while the apertures remain immobile.

Such an arrangement is also advantageous in essence, since it increases the angle of deviation and, consequently, the dispersion, almost twofold. The condition \(D = 2u\) is preserved in this case as well for every beam emerging from the instrument.

Let us find the width of the spectral interval. The expression for the angular dispersion of a focal monochromator has a somewhat different form than for a prism monochromator. In focal monochromatization the dispersion of elementary beams depends not only on \(n\), but also on their distance from the principal optical axis, since the refracting angle is a function of this distance \((h)\). If the radius of curvature is denoted by \(r\), and the distance of the point of entry (exit) of the ray from the principal optical axis by \(h\), then

\[ \sin \frac{A}{2} = \frac{h}{r}; \]

therefore

\[ \frac{dD}{d\lambda} = \frac{2\frac{h}{r}}{\sqrt{1 - n^2\left(\frac{h}{r}\right)^2}} \frac{dn}{d\lambda}, \tag{32} \]

i.e., the angular dispersion of a focal monochromator, besides \(n\),

depends on the curvature of the lens surface and the position of the point of entry (exit) of the ray relative to the principal optical axis. In the first approximation the angular dispersion is directly proportional to the distance of the point of entry (exit) of the ray from the axis \((h)\) and to the curvature of the surface \(\left(\dfrac{1}{r}\right)\). In this case the change in the denominator may be neglected, since \(\dfrac{h}{r}\) is always a proper and, moreover, small fraction.

For a focal monochromator one may distinguish two kinds of linear dispersion: transverse and longitudinal (axial). The first can be determined from formula (4). The axial dispersion characterizes the density of arrangement of the foci of monochromatic beams on the principal optical axis, or the change of the principal focal length \(f\) as a function of the wavelength, i.e.

\[ \frac{df}{d\lambda}=\frac{df}{dn}\frac{dn}{d\lambda}. \]

If \(f=f(r)\dfrac{1}{n-1}\), then

\[ df=-(f)r\,\frac{dn}{(n-1)^2}. \]

By this formula one can calculate the axial length of the spectrum; it is equal to

\[ f_{\lambda_2}-f_{\lambda_1}=\int_{n_1}^{n_2} f(r)\,\frac{dn}{(n-1)^2}. \tag{33} \]

Knowledge of the axial dispersion does not allow one to calculate the width of the spectral interval directly, since the plane of the aperture is perpendicular to the plane of the spectrum. Therefore we shall proceed as follows.

Fig. 29. Passage of monochromatic beams through the aperture of a monochromator.

Let us consider the conditions for the passage through the exit aperture of rays incident on the lens parallel to the principal optical axis. Of greatest interest are the rays emerging from the edge of the screen, since for them both the angular dispersion and the aperture angle are smallest, and therefore \(\Delta \lambda\) will have the greatest value.

Let us denote the radius of the screen by \(h_0\), and the radius of the aperture by \(a\). As is seen from Fig. 29, the interval \(\Delta \lambda\) is enclosed between \(\lambda'\) and \(\lambda''\), for which the principal focal lengths are denoted by \(f_{\lambda'}\) and \(f_{\lambda''}\).

respectively. Let us first find the width of the spectral interval on the side of the shorter wavelengths, i.e., between \(\lambda\) and \(\lambda'\). It is equal to

\[ \frac{a}{h_0}=\frac{n'-n}{n-1}. \]

In exactly the same way, the expression for the interval on the side of the longer wavelengths is

\[ \frac{a}{h_0}=\frac{n-n''}{n-1}. \]

Finally, for the whole spectral interval, for which the refractive index varies from \(n'\) to \(n''\), we obtain the expression

\[ \frac{n'-n''}{n-1}=\frac{2a}{h_0} \]

or

\[ \Delta n=\frac{2a}{h_0}(n-1). \]

It is now no longer difficult to find \(\Delta \lambda\):

\[ \Delta\lambda=\frac{2a}{h_0}(n-1)\frac{d\lambda}{dn}. \tag{34} \]

Condition (34) is valid for one lens of a focal monochromator. For two lenses, as in our case, \(\Delta\lambda\) is reduced by half, i.e.,

\[ \Delta\lambda=\frac{a}{h_0}(n-1)\frac{d\lambda}{dn}. \tag{34a} \]

Thus, the spectral width of the light beam emerging from the aperture of a focal monochromator is directly proportional to the dispersion of the lens material \(\left(\frac{d\lambda}{dn}\right)\), to the aperture radius \((a)\), and inversely proportional to the radius of the screen \((h_0)\), or to the distance of the point of exit (entry) of the rays from the principal optical axis; it increases with increasing refractive index for the mean wavelength \(\lambda\).

It follows from this that, in order to reduce \(\Delta\lambda\), it is necessary to enlarge the screen blocking the central part of the beam, i.e., to work on the peripheral part of the lens. But since enlarging the screen leads to a reduction of the luminous aperture of the lens and, consequently, to a reduction of the beam energy, lenses with a large aperture must be used.

Formula (34a) does not include the parameters that determine the optical power of the lens. Consequently, for given \(a\) and \(h\), the width of the spectral interval does not depend on the focal length.

lenses. This essentially distinguishes the focal monochromator from the prism monochromator, in which, at the cost of loss in luminous intensity (increase of \(f\)), one can increase the linear dispersion, i.e., decrease \(\Delta\lambda\) *). We judged the independence of \(\Delta\lambda\) from \(f\) from the form of expression (34a). However, it can be shown that the transverse linear dispersion, determined by formula (4), is invariant with respect to \(f\) in the case of a focal monochromator. In other words, for a given entrance aperture, increasing the focal length does not lead to a decrease of \(\Delta\lambda\) of the emergent beam.

The width \(\Delta\lambda\) calculated by formula (34a) includes all wavelengths passing through the given aperture \(a\), irrespective of their intensity.

Fig. 30. Limitation of the brightness of monochromatic beams by a screen and an aperture.

In reality, the intensity of the extreme wavelengths will be considerably weakened. Let us determine what the width of the spectral interval \(\Delta\lambda_0\) will be within which the rays of all wavelengths have relative intensities equal to 1.

For this purpose we shall assume that \(\Phi_\lambda\) is proportional to the area cut out by the aperture \(a\) from the base of the hollow cone of rays (Fig. 30). In this case the relative intensity will be equal to \(\dfrac{\Delta S}{S}\), where \(\Delta S\) is the area of the zone cut out by the aperture, and \(S\) is the area of the entire ring bounded by the rays of the extreme wavelengths. It is easy to see that

\[ \frac{\Phi}{\Phi_0} = \frac{\Delta S}{S} = \frac{a^2-x^2}{x^2\left(\dfrac{H^2}{h^2}-1\right)}. \]

For \(x=a\), \(\dfrac{\Delta S}{S}=0\); for \(x_1=a\), \(\dfrac{\Delta S}{S}=1\).

Substituting the values

\[ x=h\frac{\Delta n_0}{n-1} \quad \text{and} \quad a=h\frac{\Delta n}{n-1}, \]

we obtain:

\[ \frac{\Phi}{\Phi_0} = \frac{\dfrac{\Delta n^2}{\Delta n_0^2}-1}{\dfrac{H^2}{h^2}-1}. \]

*) Decreasing \(\Delta\lambda\) in a focal monochromator by increasing the screen also leads to a decrease in luminous intensity, but in another way.

This ratio will be equal to unity under the condition

\[ \frac{\Delta n_0}{\Delta n}=\frac{h}{H} \]

or

\[ \Delta n_0=\Delta n\,\frac{h}{H}. \]

Substituting \(\Delta n_0\) into (34a) instead of \(\Delta n\), we find

\[ \Delta\lambda_0=\frac{a}{H}(n-1)\frac{d\lambda}{dn}, \tag{35} \]

whence, by comparison with (34a), we have:

\[ \Delta\lambda_0=\frac{h}{H}\,\Delta\lambda. \tag{36} \]

From the expression obtained it is seen that the closer the quantity \(h\) is to \(H\), the greater the fraction of the rays from \(\Delta\lambda\) has relative intensity equal to unity. Graphically this may be represented by a trapezoid whose area is equal to the integral intensity of the beam emerging from the aperture of the instrument. The upper base of the trapezoid corresponds to the width \(\Delta\lambda_0\), the lower to \(\Delta\lambda\), and the height of the trapezoid corresponds to the relative intensity \(\Phi/\Phi_0\). From formulas (34) and (36) it follows that increasing \(h\) leads not only to a decrease of \(\Delta\lambda\), but also to a better distribution of the energy in the beam.

Let us calculate the magnitude of the luminous flux for the given spectral interval. The elementary luminous flux is the flux passing through an elementary zone of the light aperture of the lens, the area of which is equal to

\[ dS=2\pi h\,dh. \]

Then, considering the luminous flux proportional to \(\Delta\lambda\), we find

\[ d\Phi=B_\lambda\Delta\lambda\,dS \]

or, taking (34a) into account,

\[ \Phi_{\Delta\lambda}=2\pi B_\lambda a(n-1)\frac{d\lambda}{dn}(H-h_0). \tag{37} \]

Here \(B_\lambda\) is the brightness of the monochromatic beam for the mean wavelength.

But we saw above that not all elementary beams pass through the exit aperture completely. Let us find an expression for the luminous flux, similar to (37), for \(\Delta\lambda_0\). It is equal to:

\[ \Phi_{\Delta\lambda_0}=\pi B_\lambda\frac{a}{H}(n-1)\frac{d\lambda}{dn}(H^2-h_0^2). \tag{38} \]

From formulas (37) and (38) one can find the fraction of energy falling in the interval \(\Delta\lambda_0\). It is equal to:

\[ \frac{\Phi_{\Delta\lambda_0}}{\Phi_{\Delta\lambda}}=\frac{H+h_0}{2H}. \tag{39} \]

In our monochromator, with \(2H=50\) mm and \(h_0=15\) mm, the luminous flux for \(\Delta\lambda_0\) amounts to 0.8 of the total luminous flux.

In all the preceding reasoning it was assumed that the entire flux (within \(\Delta\lambda\)) passing through the first objective also passes through the second. But the magnitude \(\Phi_{\lambda_0}\), like \(\Phi_{\Delta\lambda}\), will also depend on the distance between the objectives.

When the objectives are separated by a distance \(l\), vignetting of the beam occurs, and precisely of those rays which are not parallel to the optical axis.

It is easy to see that the larger the part of the spectral interval subjected to vignetting, the farther apart the objectives are.

Thus, in those cases where the overall dimensions of the instrument are not limited by the operating conditions of the focal monochromator, separating the objectives may be used as an additional method of reducing the width of the spectral interval.

Fig. 31. General view of the focal monochromator (Toporets).

One may also indicate another method for improving the spectral purity of the beam emerging from the instrument.

The cross-section of the beam (Fig. 30) has the form of a ring consisting of separate monochromatic zones, the long-wavelength region being located inside and the short-wavelength region outside. By covering the outer or inner parts of the ring, one can narrow \(\Delta\lambda\); in doing so its center of gravity will shift—in the first case toward the long waves, and in the second toward the short ones.

The principal diagram of the instrument is shown in Fig. 31. Its design data are as follows. The diameter of the objectives is 50 mm,

The diameter of the screen covering the central part of the beam is 30 mm. To protect the sodium-chloride lens from the action of atmospheric moisture, it is placed in the middle of the mount: on one side of it there is a lithium-fluoride lens, and on the other a thin plane-parallel quartz plate. The gaps in the mount are sealed with lacquer.

The axial length of the spectrum in the interval from 199 mμ to 346 mμ is 61 mm. The residual transverse aberrations in the most unfavorable case give a spot 0.1 mm in diameter.

Fig. 32. Dependence of the spectral interval on wavelength.

The width of the spectral interval isolated by the focal monochromator in various regions of the spectrum with an aperture of 0.5 mm is shown in Fig. 32. From the data of the figure it is evident that in the short-wave ultraviolet, in degree of monochromatization, the instrument is almost not inferior to a prism monochromator, while in the long-wave region it surpasses absorption light filters.

Devin[^31] proposed an autocollimation system of a focal monochromator. The features of this arrangement are evident from Fig. 33. The system is a combination of a concave lens with a concave mirror. The center and radii of curvature of the refracting and reflecting

Fig. 33. Scheme of an autocollimation focal monochromator.

surfaces are denoted by \(C_1\) and \(C_2\), \(r\) and \(R\), respectively. The distance between the centers is equal to

\[ C_1C_2=\frac{r}{n_0}. \]

Suppose that a white ray emerges from point \(F\). At the point of incidence the ray is refracted and is decomposed into elementary monochromatic rays. Among the entire set of rays there will be one which, after being refracted in the lens, falls normally on the mirror and, after reflection, returns along the old path to point \(F\). If the exit slit \(F'\) is placed next to the entrance slit, and the system is tilted accordingly, then such a device will constitute an autocollimation focal monochromator. To pass from one region of the spectrum to another, it is necessary to move the optical system parallel to itself relative to the slits.

Figure 34 shows a double focal monochromator. It uses the scheme first proposed by Shcherniuk \(^{32}\). The systems \(L_1\) and \(L_2\) are two half-lenses with reflecting surfaces; the straight lines \(c_1C_1F_1\) and \(c_2C_2F_2\) are their optical axes, and \(F_e\) and \(F_s\) are the entrance and exit slits. The systems \(L_1\) and \(L_2\) give a spectral image of the corresponding slits on the line \(AA\), which coincides with the axis of symmetry of the monochromator. The middle slit is the gap between the screen \(L\) and the mirror \(M\). By moving this slit along the axis \(AA\), different spectral regions can be obtained at the exit slit.

Fig. 34. Diagram of a double focal monochromator.

The use of focal monochromators for isolating individual lines makes it possible to obtain beams several times more powerful than with ordinary prismatic monochromators \(^{28}\).

In conclusion it should be said that focal monochromators undoubtedly deserve attention. Although in degree of monochromatization they are inferior to ordinary prismatic instruments, in many cases they can be used as narrow-band filters, especially in those regions of the spectrum for which filters do not exist or where their selection is very limited.

CONCLUSION

In our review we have considered the basic principles of the design of monochromators and have only incidentally indicated the merits and shortcomings of individual designs. We did not set ourselves the goal of making a comparative evaluation of existing models in order to say which design is the best. Such an evaluation would require a special experimental investigation of various models according to a definite program. And even if such an investigation were available, it would hardly be possible to settle on any single design, since it is impossible to create a universal instrument suitable for different regions of the spectrum and for solving a wide range of problems. The requirements imposed on an instrument follow from the specific conditions of the problem being solved, and, along with the technical and experimental qualities of the instrument, one must always also take into account the economic factor—the cost of the instrument. Therefore, alongside complex and expensive instruments of a high degree of monochromatization, simplified ones must also exist.

Improvement in the quality of monochromators should evidently proceed, first, along the line of using better projection systems: namely, the use of aspherical and coated optics, as well as obtaining mirrors with a high reflection coefficient. Secondly, along the line of using substances with high dispersion, and also the use of diffraction gratings.

For the reasons stated above, it may be expected that monochromators with catoptric systems will become the most widespread.

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