# Dispersive Prisms and Gratings

This is a continuation from the previous tutorial - LDPC-coded differential modulation decoding algorithms.

### Introduction

This category of measurement systems usually consists of those in which separation of the radiation into its spectral components, or dispersion, is accomplished by the use of an optical element possessing a known functional dependence on wavelength—specifically, prisms and diffraction gratings.

### Glossary

• $$A_p$$ prism angle
• $$B$$ prism base
• $$D_p$$ angle of minimum derivation
• $$d$$ grating constant
• $$E$$ irradiance
• $$N$$ number of slits
• $$n$$ refractive index
• $$p$$ order number
• $$\text{RP}$$ resolving power
• $$r$$ angles
• $$W$$ prism width
• $$\beta$$ angle
• $$\gamma$$ angle

### Prisms

The wavelength dependence of the index of refraction is used in prism spectrometers. Such an optical element disperses parallel rays or collimated radiation into different angles from the prism according to wavelength.

Distortion of the image of the entrance slit is minimized by the use of plane wave illumination. Even with plate wave illumination, the image of the slit is curved because not all of the rays from the entrance slit can traverse the prism in its principal plane. A prism is shown in the position of minimum angular deviation of the incoming rays in Fig. 2.

At minimum angular deviation, maximum power can pass through the prism. For a prism adjusted to the position of minimum deviation,

$\tag{1}r_1=r_2=A_p/2$

and

$\tag{2}i_1=i_2=[D_p+A_p]/2$

where

• $$D_p$$ = angle of minimum deviation for the prism
• $$A_p$$ = angle of the prism
• $$r_1$$ and $$r_2$$ = internal angles of refraction
• $$i_1$$ and $$i_2$$ = angles of entry and exit

The angle of minimum deviation $$D_p$$ varies with wavelength. The angular dispersion is defined as $$dD_p/d\lambda$$, while the linear dispersion is

$\tag{3}\frac{dx}{d\lambda}=F\frac{dD_p}{d\lambda}$

where $$F$$ is the focal length of the camera or imaging lens and $$x$$ is the distance across the image plane. It can be shown that

$\tag{4}\frac{dD_p}{d\lambda}=\frac{B}{W}\frac{dn}{d\lambda}$

where

• $$B$$ = base length of the prism
• $$W$$ = width of the illumination beam
• $$n$$ = index of refraction
• $$\frac{dx}{d\lambda}=F\frac{B}{W}\frac{dn}{d\lambda}$$

The resolving power $$\text{RP}$$ of an instrument may be defined as the smallest resolvable wavelength difference, according to the Rayleigh criterion, divided into the average wavelength in that spectral region. The limiting resolution is set by diffraction due to the finite beam width, or effective aperture of the prism, which is rectangular. Thus,

$\tag{5}\text{RP}_p=B\frac{dn}{d\lambda}$

If the entire prism face is not illuminated, then only the illuminated base length must be used for $$B$$.

### Gratings

A grating is an $$n$$-slit system used in Fraunhofer diffraction with interference arising from division of the incident, plane wave front. Thus it is a multiple beam interferometer.

$\tag{6}p\lambda=d(\sin\theta+\sin\phi)$

where

• $$p$$ = order number (= 0, 1, 2, . . . ) of the principal maxima
• $$d$$ = the grating constant or spacing (the distance between adjacent slits)
• $$\phi$$ = angle of incidence
• $$\theta$$ = angle of diffraction
• $$w$$ = width of any one slit

The most common case is $$\phi=0$$, so that

$\tag{7}p\lambda=d\sin\theta$

$\tag{8}\begin{split}E=&E_0\{\sin((\pi{w}\sin\theta)/\lambda)/((\pi{w}\sin\theta)/\lambda)\}^2\\&\times\{\sin((N\pi{d}\sin\theta)/\lambda)/\sin((\pi{d}\sin\theta)/\lambda)\}^2\end{split}$

where $$N$$ is the number of slits or grooves. This equation is more often written as:

$\tag{9}E=E_0[(\sin\beta)/\beta]^2[(\sin{N\gamma})/\sin{\gamma}]^2$

which can be considered to be

\tag{10}\begin{align}E=&(\text{constant})\times(\text{single-slit diffraction function})\\&\times(N\text{-slit interference function})\end{align}

These considerations are for unblazed gratings. For a diffraction grating, the angular dispersion is given (for angle $$\phi$$ constant) by

$\tag{11}\frac{dD_g}{d\lambda}\qquad\text{or}\qquad\frac{d\theta}{d\lambda}=\frac{p}{d\cos\theta}$

The resolving power is given by

$\tag{12}\text{RP}_g=pN$

### Prism and Grating Configuration and Instruments

#### Classical

There are several basic prism and grating configurations and spectrometer designs which continue to be useful. One of the oldest spectrometer configurations is shown in Fig. 3. Reflective interactions and prism combinations are used in Figs. 4, 5, and 6.

Dispersion without deviation is realized in Figs. 7 and 8, while half-prisms are used in Fig. 9 in an arrangement which uses smaller prisms but still attains the same beam width.

A few classical prism instrumental configurations are shown in Figs. 10, 11, and 12. Multiple-pass prism configurations are illustrated in Figs. 13 and 14.

A well-known example of a single beam double-pass prism infrared spectrometer was the Perkin-Elmer Model 112 instrument shown in Fig. 15.

Infrared radiation from a source is focused by mirrors $$M_1$$ and $$M_2$$ on the entrance slit $$S_1$$ of the monochromator. The radiation beam from $$S_1$$, path 1, is collimated by the off-axis paraboloid $$M_3$$ and a parallel beam traverses the prism for a first refraction.

The beam is reflected by the Littrow mirror $$M_4$$, through the prism for a second refraction, and focused by the paraboloid, path 2, at the corner mirror $$M_6$$.\ The radiation returns along path 3, traverses the prism again, and is returned back along path 4 for reflection by mirror $$M_7$$ to the exit slit $$S_2$$.

By this double dispersion, the radiation is spread out along the plane of $$S_2$$. The radiation of the frequency interval which passes through $$S_2$$ is focused by mirrors $$M_8$$ and $$M_9$$ on the thermocouple TC.

The beam is chopped by CH, near $$M_6$$, to produce a voltage (at the thermocouple) which is proportional to the radiant power or intensity of the beam. This voltage is amplified and recorded by an electronic potentiometer. Motor-driven rotation of Littrow mirror $$M_4$$ causes the infrared spectrum to pass across exit slit $$S_2$$ permitting measurement of the radiant intensity of successive frequencies.

Gratings can be used either in transmission or reflection. Another interesting variation comes from their use in plane or concave reflection form. The last was treated most completely by Rowland, who achieved a useful combination of focusing and grating action.

He showed that the radius of curvature of the grating surface is the diameter of a circle (called the Rowland circle). Any source placed on the circle will be imaged on the circle, with dispersion, if the rulings are made so that $$d$$ is constant on the secant to the grating-blank (spherical) surface.

The astigmatism acts so that a point source on a Rowland circle is imaged as a vertical line perpendicular to the plane of the circle. Rowland invented and constructed the first concave grating mounting, illustrated in Fig. 16.

If dispersion is sufficiently large, one may find overlapping of the lines from one order with members of the spectra belonging to a neighboring order. Errors and imperfections in the ruling of gratings can produce spurious images which are called ‘‘ghosts.’’

Also, the grooves in a grating can be shaped so as to send more radiation along a preferred direction corresponding to an order other than the zero order. Such gratings are said to be blazed in that order. These issues and many more involved in the production of gratings by ruling engines were thoroughly discussed by Harrison in his 1973 paper ‘‘The Diffraction Grating—An Opinionated Appraisal.’’

Six more grating configurations which are considered to be ‘‘classics’’ are:

1. Paschen-Runge, illustrated in Fig. 17. In this argument, one or more fixed slits are placed to give an angle of incidence suitable for the uses of the instrument. The spectra are focused along the Rowland circle $$P P'$$, and photographic plates, or other detectors, are placed along a large portion of this circle.

2. Eagle, shown in Fig. 18. This is similar to the Littrow prism spectrograph. The slit and plate holder are mounted close together on one end of a rigid bar with the concave grating mounted on the other end.

3. Wadsworth, shown in Fig. 19. The Rowland circle is not used in this mounting in which the grating receives parallel light.

4. Ebert-Fastie, shown in Fig. 20. The Ebert-Fastie features a single, spherical, collimating mirror and a grating placed symmetrically between the two slits. The major advantage of the Ebert system is the fact that it is self-correcting for spherical aberration. With the use of curved slits , astigmatism is almost completely overcome.

5. Littrow, shown in Fig. 10. The Littrow system has slits on the same side of the grating to minimize astigmatism. An advantage of the Littrow mount, therefore, is that straight slits can be used. In fact, such slits may be used even for a spherical collimating mirror if the aperture is not too large. Its greatest disadvantage is that it does not correct for spherical aberration—not too serious a defect for long focal-length/small-aperture instruments. If an off-axis parabola is used to collimate the light, aberrations are greatly reduced.

6. Pfund, shown in Figs. 12 and 21. This is an on-axis, Pfund-type grating instrument. Incident infrared radiation, focused by a collimating lens on the entrance slit and modulated by a chopper, passes through the central aperture of plane mirror $$M_1$$. Reflected by the paraboloidal mirror $$P_1$$, it emerges as a parallel beam of radiation, which is reflected by mirror $$M_1$$ to the grating. The grating is accurately located on a turntable, which may be rotated to scan the spectrum. From the grating, the diffracted beam, reflected by mirror $$M_2$$, is focused by a second paraboloid $$P_2$$ through the central aperture of mirror $$M_2$$ to the exit slit. The emerging beam is then focused by the ellipsoidal mirror $$M_3$$ on the detector.

An off-axis, double-pass grating instrument is illustrated in Fig. 22.

Combinations of prisms and gratings are not uncommon. An illustrative and complex prism-grating, double-monochromator spectrometer designed by Unicam Instruments, Ltd. is shown in Fig. 23.

The prism monochromator had four interchangeable prisms, and the grating monochromator had two interchangeable gratings. The two monochromators, ganged by cams which are linear in wave number, were driven by a common shaft. The instrument could be used either as a prism-grating double monochromator, or as a prism spectrometer by blanking the grating monochromator. Gratings, prisms, and cams could be automatically interchanged by means of push buttons. Magnetically operated slits, programmed by a taped potentiometer, provided a constant energy background. A star-wheel, time-sharing, beam attenuator was used in the double-beam photometer.

#### Comtemporary

In recent years there has been more attention paid to total system design and integration for specific purposes and applications, as for analytical atomic and molecular spectroscopy in analytical chemistry.

Thus the conventional dispersive elements are often used in the classical configurations with variations. Innovations have come especially in designs tailored for complete computer control; introduction of one- and two-dimensional detector arrays as well as new detector types (especially for signal matching); the use of holographic optical elements either alone or combined with holographic gratings; and special data-processing software packages, displays, and data storage systems.

Some examples found by a brief look through manufacturers’ literature and journals such as Spectroscopy, Physics Today, Laser Focus, Photonics Spectra, and Lasers & Optronics, are presented in Table 1. Most of these systems are designed for analytical spectroscopy.

The next tutorial gives a detailed introduction of integrated optics.