It is fundamental to the nature of diffraction gratings that errors are relatively easy to measure, although not all attributes are equally detectable or sometimes even definable.

For example, a grating with low background (in the form of scatter or satellites) can be simply tested for Rowland ghosts on an optical bench. With a mercury lamp or a laser source, and a scanning slit connected to a detector and recorder, a ghost having intensity 0.002% of the intensity of the main line can be easily observed. The periodic error in the groove spacing giving rise to such a ghost may be less than one nanometer.

Grating ghosts are measured at Newport by making the grating part of a scanning spectrometer and illuminating it with monochromatic light, such as that from a mercury isotope lamp (isotope 198 or 202) or a helium-neon laser. On scanning both sides of the parent line, using a chart recorder and calibrated attenuators, it is easy to identify all ghost lines and to measure their intensities relative to the parent line.

The importance of ghosts in grating applications varies considerably. In most spectrophotometers, and in work with low-intensity sources, ghosts play a negligible role. In Raman spectroscopy, however, even the weakest ghost may appear to be a Raman line, especially when investigating solid samples, and hence these ghosts must be suppressed to truly negligible values.

Ghosts are usually classified as Rowland ghosts, Lyman ghosts and satellites.

Rowland ghosts are spurious lines seen in some grating spectra that result from large-scale (millimeter) periodic errors in the spacing of the grooves (see Figure 11-1). These lines are usually located symmetrically with respect to each strong spectral line at a (spectral) distance from it that depends on the period of the error, and with an intensity that depends on the amplitude of this error.

Figure 11-1.   'Ghost' trace showing Rowland ghosts caused by the periodic error of 2.54 mm in the lead screw of the Newport MIT 'B' engine. MR215 is an echelle grating, with 52.67 g/mm, in this case tested in the 54^th order.

If the curve of groove spacing error vs. position is not simply sinusoidal, there will be a number of ghosts on each side of the parent line representing the various orders from each of the harmonics of the error curve. On engines with mechanical drives, Rowland ghosts are associated primarily with errors in the lead or pitch of the precision screw, or with the bearings of the ruling engine. As a consequence, their location depends upon the number of grooves ruled for each complete turn of the screw. For example, if the ruling engine has a pitch of 2 mm, and a ruling is made at 1200 grooves/mm, 2400 grooves will be ruled per turn of the screw, and the ghosts in the first order can be expected to lie at Dl = ± l/2400 from the parent line l, with additional ghosts located at integral multiples of Dl. In gratings ruled on engines with interferometric feedback correction mechanisms, Rowland ghosts are usually much less intense, but they can arise from the mechanisms used in the correction system if care is not taken to prevent their occurrence.

If the character of the periodic errors in a ruling engine were simply harmonic, which is rarely true in practice, the ratio of the diffracted intensities of the first order Rowland ghost (I_{RG (m=1)}) to that of the parent line (I_PL) is

,

(11-1)

where A is the peak simple harmonic error amplitude, a is the angle of incidence, and l is the diffracted wavelength.

The second-order Rowland ghost I_{RG (m=2)} will be much less intense (note the exponent):

.

(11-2)

Higher-order Rowland ghosts would be virtually invisible. The ghost intensity is independent of the diffraction order m of the parent line, and of the groove spacing d. In the Littrow configuration, Eq. (11-1) becomes

,     in Littrow,

(11-3)

an expression derived in 1893 by Rowland.

These simple mathematical formulas do not always apply in practice when describing higher-order ghost intensities, since the harmonic content of actual error curves gives rise to complex amplitudes that must be added vectorially and then squared to obtain intensity functions. Fortunately, the result of this complication is that ghost intensities are generally smaller than those predicted from the peak error amplitude.

The order of magnitude of the fundamental harmonic error amplitude can be derived from Eq. (11-1) [or Eq. (11-3)]. For example, a 1200 g/mm grating used in the m = 1 order in Littrow will show a 0.14% first-order ghost intensity, compared with the parent line, for a fundamental harmonic error amplitude of A = 10 nm. For some applications, this ghost intensity is unacceptably high, which illustrates the importance of minimizing periodic errors of ruling. For Raman gratings and echelles, the amplitude A of the periodic error must not exceed one nanometer; the fact that this has been accomplished is a remarkable achievement.

Ghost lines observed at large spectral distances from their parent lines are called Lyman ghosts. They result from compounded periodic errors in the spacing of the grating grooves; the period of Lyman ghosts is on the order of a few times the groove spacing.

Lyman ghosts can be said to be in fractional-order positions (see Figure 11 2). Thus, if every other groove is misplaced so that the period contains just two grooves, ghosts are seen in the half-order positions. The number of grooves per period determines the fractional-order position of Lyman ghosts.

Usually it is possible to find the origin of the error in the ruling engine once its periodicity is determined. It is important that Lyman ghosts be kept to a minimum, because they are not nearly as easy to identify as Rowland ghosts.

Satellites are false spectral lines usually occurring very close to the parent line. Individual gratings vary greatly in the number and intensity of satellites which they produce. In a poor grating, they give rise to much scattered light, referred to as grass (so called since this low intensity scattered light appears like a strip of lawn when viewed with green mercury light). In contrast to Rowland ghosts, which usually arise from errors extending over large areas of the grating, each satellite usually originates from a small number of randomly misplaced grooves in a localized part of the grating. With laser illumination, a relative background intensity of 10^-7 is easily observable with the eye.

Figure 11-2.   'Ghost' trace showing Lyman ghosts (the small spikes between orders 2 and 3), which can be associated with fractional order positions, e.g., an error every five grooves corresponds to a fraction order of 1/5.

Grating efficiency measurements are generally performed with a double monochromator system. The first monochromator supplies monochromatic light derived from a tungsten lamp, mercury arc, or deuterium lamp, depending on the spectral region involved. The grating being tested serves as the dispersing element in the second monochromator. In the normal mode of operation, the output is compared with that from a high-grade mirror coated with the same material as the grating. The efficiency of the grating relative to that of the mirror is reported (relative efficiency), although absolute efficiency values can also be obtained (either by direct measurement or through knowledge of the variation of mirror reflectance with wavelength). For plane reflection gratings, the wavelength region covered is usually 190 nm to 2.5 µm; gratings blazed farther into the infrared are often measured in higher orders. Concave reflection gratings focus as well as disperse the light, so the entrance and exit slits of the second monochromator are placed at the positions for which the grating was designed (that is, concave grating efficiencies are measured in the geometry in which the gratings are to be used). Transmission gratings are tested on the same equipment, with values given as the ratio of diffracted intensity to the intensity falling directly on the detector from the light source (i.e., absolute efficiency).

Curves of efficiency vs. wavelength for plane gratings are made routinely on all new master gratings produced by Newport, both plane and concave, with light polarized in the S and P planes to assess the presence and amplitudes (if any) of anomalies. Such curves are furnished by Newport upon request (for an example, see Figure 11-3).

Figure 11-3.   Example of an efficiency curve. This efficiency curve is specific to the particular grating under test, as well as the conditions of illumination (the incidence and diffraction angles).

One of the most critical tests an optical system can undergo is the Foucault knife-edge test. This test not only reveals a great deal about wavefront deficiencies but also locates specific areas (or zones) on the optical component where they originate. The test is suited equally well to plane and concave gratings (for the former, the use of very high grade collimating optics is required). The sharper (i.e., more abrupt) its knife-edge cut-off, the more likely that a grating will yield high resolution.

The sensitivity of the test depends on the radius of the concave grating (or the focal length of the collimating system), and may exceed that of interferometric testing, although the latter is more quantitative.

The Foucault test is a sensitive and powerful tool, but experience is required to interpret each effect that it makes evident. All Newport master plane gratings, large plane replicas and large-radius concave gratings are checked by this method (see Figure 11-4).

Figure 11-4.   A grating under test on the Newport 5-meter test bench. Light from a mercury source (not shown, about 5 meters to the right) is collimated by the lens (shown) which illuminates the grating (shown on a rotation stage); the same lens refocuses the diffracted light to a plane very near the light source, where the diffracted wavefront can be inspected.

Any departure from perfect flatness of the surface of a plane grating, or from a perfect sphere of the surface of a concave grating, as well as variations in the groove spacing, depth or parallelism, will result in a diffracted wavefront that is less than perfect. In order to maintain resolution, this departure from perfection is generally held to l/4 or less, where l is the wavelength of the light used in the test. To obtain an understanding of the magnitudes involved, it is necessary to consider the angle at which the grating is used. For simplicity, consider this to be the blaze angle, under Littrow conditions. Any surface figure error of height h will cause a wavefront deformation of 2h cosq, which decreases with increasing |q |. On the other hand, a groove position error p introduces a wavefront error of 2p sinq, which explains why ruling parameters are more critical for gratings used in high-angle configurations.

A plane grating may produce a slightly cylindrical wavefront if the groove spacing changes linearly, or if the surface figure is similarly deformed. In this special case, resolution is maintained, but focal distance will vary with wavelength.

Wavefront testing can be done conveniently by mounting a grating at its autocollimating angle (Littrow) in a Twyman-Green interferometer or a phase measuring interferometer (PMI; see Figure 11-5). Newport interferometers have apertures up to 150 mm (6 inches). With coherent laser light sources, however, it is possible to make the same measurements with a much simpler Fizeau interferometer, equipped with computer fringe analysis.

It should be noted that testing the reflected wavefront – that is, illuminating the grating in zero order – is generally inadequate since this arrangement will examine the flatness of the grating surface but tells nothing about the uniformity of the groove pattern.

Periodic errors give rise to zig-zag fringe displacements. A sudden change in groove position gives rise to a step in the fringe pattern; in the spectrum, this is likely to appear as a satellite. Curved fringes due to progressive ruling error can be distinguished from figure problems by observing fringes obtained in zero, first and higher orders. Fanning error (non-parallel grooves) will cause spreading fringes. Figure 11-6a shows a typical interferogram, for an echelle grating measured in Littrow in the diffraction order of use (m = 33).

Figure 11-5.   A plane grating under test on a phase measuring interferometer. The grating is tested in the Littrow configuration so that the flatness of the diffracted wavefront is evaluated.

Experience has shown that the sensitivity of standard interferograms for grating deficiencies equals or exceeds that of other plane grating testing methods only for gratings used at high angles. This is why the interferometric test is especially appropriate for the testing of echelles and other gratings used in high diffraction orders.

Figure 11-6a.   Example of an interferogram and histogram generated by a Phase Measuring Interferometer (PMI). In this example, a grating is illuminated in a circular region 50 mm in diameter, and its diffracted wavefront at l = 632.8 nm in the m = 33 order is recorded.

Resolving power (defined in Section 2.4) is an crucial characteristic of diffraction gratings since it is a measure of the fundamental property for which gratings are used: it quantifies the ability of the grating (when used in an optical system) to separate two nearby wavelengths. Often resolving power is specified to be great enough that the resolution of the optical system will be slit limited rather than grating limited (see Section 8.3).

Resolving power is generally measured in a spectrometer with a large focal length and very narrow slits in which the light source has fine spectral structure; an example is the hyperfine spectrum of natural Hg near 546.1 nm (see Figure 11-6). The spectral lines are identified, and the wavelengths of those that are distinguishable ('resolvable') are subtracted, and this difference Dl is divided into l = 546.1 nm according to Eq. (2-17); the smaller the wavelength difference, the greater the resolving power.

Figure 11-6b.   Hyperfine structure of natural Hg near 546.1 nm. . Several spectral lines are identified. Visual identification of two distinct emission lines centered on l and separated by Dl implies a resolving power at least as great as l/Dl.

Resolving power is measured on Newport diffraction gratings using a specially-designed Czerny-Turner spectrograph, whose concave mirrors have very long focal lengths (10 m) so that very large astronomical gratings may be tested (see Section 13.3).

As discussed in Chapters 2 and 10, light that leaves a grating surface that does not follow the grating equation (2-1) is called scattered light. Scattered light is generally measured in one of two ways: either with cut-off filters (which absorb one part of the spectrum while transmitting the other part) or by using monochromatic light (from an atomic emission source or a laser, or by the use of interference filters that transmit a narrow spectral range).

Newport has two specially-designed instruments to measure light scattered from small regions on the surface of a mirror or grating: one uses red HeNe light (l = 632.8 nm) to illuminate the grating, and the uses a Hg source to illuminate the grating (the light reaching the detector is filtered to transmit a narrow spectral band around 254 nm). These "scatter checkers" provide several degrees of freedom so that light scattered between diffraction orders (called inter-order scatter) can be attributed to areas on the grating surface.

Figure 11-7 shows a simplified schematic diagram of the scatter checker. The beam from a polarized HeNe laser is spatially filtered to remove speckle and is then directed onto a concave focusing mirror that brings the beam to focus at the detector plane. The detector is a photomultiplier that, in combination with a programmable-gain current amplifier, provides eight decades of dynamic range. A PC equipped with a data acquisition card is used to process and store the detector signal.

Scatter measurements are made by first obtaining a reference beam profile (see Figure 11-8), or "instrument signature," by translating the test optic out of the way and rotating the detector through the beam in incremental steps over a predetermined angular range. The test optic is then translated into the beam path and the detector passed through the reflected (or diffracted) beam from the test optic over the same angular range used to make the reference measurement. The sample and reference beam profiles are "mirror images" of one another, so it is necessary to invert one before a comparison is made. Any difference between the sample and reference beam profiles can be attributed to light scattered from the optic under test.

Figure 11-7.   Schematic of the Newport red HeNe scatter measuring apparatus. In order to minimize the effects that other diffraction orders may have on the scattered light readings, this instrument is not enclosed so that any light that leaves the grating in a direction other than toward the detector will travel a long distance before encountering a reflecting or scattering surface.

Figure 11-8.   Typical plot of data obtained from the Newport red HeNe scatter measuring instrument. This plot of the measured signal vs. angle of rotation of the detector (from a diffracted order) shows the reference beam profile (the "instrument signature").

It is important to apply the lessons of Chapter 10 to the interpretation of grating scatter measurements. That is, even a "perfect" grating (as defined in Section 10.1.6) illuminated with monochromatic light will cause other diffraction orders to propagate, and some of this light energy may reach the detector of the scatter measuring apparatus. This is important when comparing the scatter characteristics of a grating with those of a high-quality mirror (using the latter as a reference'); the mirror produces only the m = 0 order (specular reflection) and will therefore exhibit lower scatter than even a "perfect" grating. This subtle point must be considered in defining the instrument signature of a grating-based optical system by using a mirror.

In analyzing grating scatter measurements, care must be taken to account for any stray light that is due to the measurement apparatus rather than the grating, as discussed in Sections 10.2 and 10.3.

The consequence of undesired energy reaching the detector in a spectrometer is a reduction in photometric accuracy, since some light reaches the detector that cannot be attributed to the transmission (or absorption) of the sample at the analytical wavelength.

Instrumental stray light, like scattered light, is generally measured either with cut-off filters or monochromatic light.

Instrumental stray light is commonly measured in by using a set of high-pass cut-off optical filters (whose transmission curves look like that in Figure 11-9). The spectrometer is then scanned toward shorter wavelengths and the transmittance measured; once the transmittance level has reached a fairly steady minimum (a plateau), this reading is taken to be the stray light.¹⁴⁰

Figure 11-9.   Transmission curve of a typical high-pass cut-off filter. A filter of this type is generally specified by the cut-off wavelength l_C, the wavelength at which its transmission coefficient is 50%. The slope of the transmission curve near l_C should be as steep as possible.

The instrument is tuned to the analytical wavelength l and a series of filters, each with a successively higher cut-off wavelength l_C (>l), is placed in the beam and intensity readings taken at the detector. [Generally l_C should exceed l by at least 20 nm, in the visible spectrum, to ensure than virtually no light of the analytical wavelength l passes through the filter and complicates the readings.] Nonzero readings indicate the presence of stray light. A proper study requires measurements at more than one analytical wavelength since stray light properties cannot be easily extrapolated (due to the different wavelength dependencies of the causes of grating scatter and instrumental stray light noted above, and – for monochromators – the fact that all rays diffracted from or scattered by the grating change direction as the grating is rotated).

Another method for measuring instrumental stray light is to replace the polychromatic light source (used with cut-off filters) with a narrow-band monochromatic light source. Atomic emission sources provide narrow spectral emission lines that can be used for this purpose; lasers can be used; and broad-spectrum sources can be used in conjunction with bandpass filters.

Kaye¹⁴¹ describes a technique in which monochromatic light is used to determine the amount of power detected at all wavelength settings for a given input wavelength; this quantity is called the slit function. The spectrometer (with slit widths w) is illuminated by light whose central wavelength is l, and whose spectral width Dl is very narrow (Dl << l). Scanning through the full wavelength range of the instrument (the wavelength setting being denoted by ; see Section 10.1.6) and recording the power at each setting yields the slit function , which we may write as

=

(11-4)

where E_l is the power emitted by the source, is the transmittance of the optical system (between the source and the detector), R_l is the sensitivity of the detector, and c is a constant of proportionality. If we had knowledge of the slit function for all input wavelengths l and for all wavelength settings , we would be able to write for any wavelength setting the following integral:

=

(11-5)

which represents the total power (for all wavelengths) recorded at wavelength setting . In practice, the bounds of integration are not 0 and ¥, but are instead determined by the spectral sensitivity limits of the detector.

Stray light can then be expressed as the ratio of the intensities (powers) of the scattered light and principal beam.¹⁴²

Often the unwanted light in a spectrometer is quantified not by instrumental stray light but by the signal-to-noise ratio (SNR), a dimensionless quantity of more relevance to instrumental specification. The SNR is defined as the ratio of the signal (the desired power incident on the detector) to the noise (the undesired power, equivalent in our definition to the instrumental stray light).

Another specification of instrumental stray light is given in absorbance, a dimensionless quantity defined by

A =

(11-6)

where T is the percent transmittance (0 £ T £ 100). Higher values of A correspond to lower transmittances, and instrumental stray light plays an important role in the highest value of A for which the readings are accurate; an instrument for which the stray light is about 1% as intense as the signal at a given wavelength cannot provide absorbance readings of any accuracy greater than A » 2.

When the stray light power s is known (as a percentage of the signal), Eq. (11-6) may be modified to be made more accurate:¹⁴³

A =

(11-7)