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9.0. INTRODUCTION [top]
Efficiency and its variation with wavelength and spectral order are important characteristics of a diffraction grating. For a reflection grating, efficiency is defined as the energy flow (power) of monochromatic light diffracted into the order being measured, relative either to the energy flow of the incident light (absolute efficiency) or to the energy flow of specular reflection from a polished mirror substrate coated with the same material (relative efficiency). [Intensity may substitute for energy flow in these definitions.] Efficiency is defined similarly for transmission gratings, except that an uncoated substrate is used in the measurement of relative efficiency.
High-efficiency gratings are desirable for several reasons. A grating with high efficiency is more useful than one with lower efficiency in measuring weak transition lines in optical spectra. A grating with high efficiency may allow the reflectivity and transmissivity specifications for the other components in the spectrometer to be relaxed. Moreover, higher diffracted energy may imply lower instrumental stray light due to other diffracted orders, as the total energy flow for a given wavelength leaving the grating is conserved (being equal to the energy flow incident on it minus any scattering and absorption).
Control over the magnitude and variation of diffracted energy with wavelength is called blazing, and it involves the manipulation of the micro-geometry of the grating grooves. As early as 1874, Lord Rayleigh recognized that the energy flow distribution (by wavelength) of a diffraction grating could be altered by modifying the shape of the grating grooves.79 It was not until four decades later that R.W. Wood showed this to be true when he ruled a grating on which he had controlled the groove shape, thereby producing the first deliberately blazed diffraction grating.80
The choice of an optimal efficiency curve for a grating depends on the specific application. For some cases the desired instrumental response is linear with wavelength; that is, the ratio of intensity of light and the electronic signal into which it is transformed is to be nearly constant across the spectrum. To approach this as closely as possible, the spectral emissivity of the light source and the spectral response of the detector should be considered, from which the desired grating efficiency curve can be derived. Usually this requires peak grating efficiency in the region of the spectrum where the detectors are least sensitive; for example, a visible-light spectrometer using a silicon detector would be much less sensitive in the blue than in the red, suggesting that the grating itself be blazed to yield a peak efficiency in the blue.
A typical efficiency curve (a plot of absolute or relative diffracted efficiency vs. diffracted wavelength l) is shown in Figure 9-1. Usually such a curve shows
Figure 9-1. A typical (simplified) efficiency curve. This curve shows the efficiency E of a grating in a given spectral order m, measured vs. the diffracted wavelength l. The peak efficiency EP occurs at the blaze wavelength lB.
a single maximum, at the peak wavelength (or blaze wavelength) lB. This curve corresponds to a given diffraction order m; the peak of the curve decreases in magnitude and shifts toward shorter wavelengths as |m| increases. The efficiency curve also depends on the angles of use (i.e., the angles of incidence and diffraction). Moreover, the curve depends on the groove spacing d (more appropriately, on the dimensionless parameter l/d) and the material with which the grating is coated (for reflection gratings) or made (for transmission gratings).
In many instances the diffracted power depends on the polarization of the incident light. P-plane or TE polarized light is polarized parallel to the grating grooves, while S-plane or TM polarized light is polarized perpendicular to the grating grooves (see Figure 9-2). For completely unpolarized incident light, the efficiency curve will be exactly halfway between the P and S efficiency curves.
Figure 9-2. S and P polarizations The P polarization components of the incident and diffracted beams are polarized parallel to the grating grooves; the S components are polarized perpendicular to the P components. Both the S and P components are perpendicular to the propagation directions.
Usually light from a single spectral order m is used in a spectroscopic instrument, so a grating with ideal efficiency characteristics would diffract all of the power incident on it into this order (for the wavelength range considered). In practice, this is never true: the distribution of the power by the grating depends in a complicated way on the groove spacing and profile, the spectral order, the wavelength, and the grating material.
Anomalies are locations on an efficiency curve (efficiency plotted vs. wavelength) at which the efficiency changes abruptly. First observed by R. W. Wood, these sharp peaks and troughs in an efficiency curve are sometimes referred to as Wood's anomalies. Anomalies are rarely observed in P polarization efficiency curves, but they are often seen in S polarization curves (see Figure 9-3). Anomalies are discussed in more detail in Section 9.13.
Figure 9-3. Anomalies in the first order for a typical grating with triangular grooves. The P efficiency curve (solid line) is smooth, but anomalies are evident in the S curve (dashed line). The passing-off locations are identified by their spectral order at the top of the figure.
9.1. GRATING EFFICIENCY AND GROOVE SHAPE [top]
The maximum efficiency of a grating is typically obtained with a simple smooth triangular groove profile, as shown in Figure 9-4, when the groove (or blaze) angle qB is such that the specular reflection angle for the angle of incidence is equal (in magnitude and opposite in sign) to the angle of diffraction (see Section 2.8). Ideally, the groove facet should be flat with smooth straight edges, and be generally free from irregularities on a scale comparable to a small fraction (< 1/10) of the wavelength of light being diffracted.
Fraunhofer was well aware that the distribution of power among the various diffraction orders depended on the shape of the individual grating grooves. Wood, many decades later, was the first to achieve a degree of control over the groove shape, thereby concentrating spectral energy into one angular region. Wood's gratings were seen to light up, or blaze, when viewed at the correct angle.
Figure 9-4. Triangular groove geometry. The angles of incidence a and diffraction b are shown in relation to the facet angle qB. GN is the grating normal and FN is the facet normal. When the facet normal bisects the angle between the incident and diffracted rays, the grating is used in the blaze condition. The blaze arrow (shown) points from GN to FN.
Gratings with triangular grooves can be generated by mechanical ruling, or by blazing sinusoidal groove profiles by ion etching. The efficiency behavior of gratings with triangular groove profiles may be divided into six families, depending on the blaze angle:81
Very low blaze angle gratings (qB < 5°) exhibit efficiency behavior that is almost perfectly scalar; that is, polarization effects are virtually nonexistent. In this region, a simple picture of blazing is applicable, in which each groove facet can be considered a simple flat mirror. The diffracted efficiency is greatest for that wavelength that is diffracted by the grating in the same direction as it would be reflected by the facets. This efficiency peak occurs in the m = 1 order at l/d = 2 sinq (provided the angle between the incident and diffracted beams is not excessive). At lB/2, where lB is the blaze wavelength, the diffracted efficiency will be virtually zero (Figure 9-5) since for this wavelength the second-order efficiency will be at its peak. Fifty-percent absolute efficiency is obtained from roughly 0.7lB to 1.8lB.
Figure 9-5. First-order theoretical efficiency curve: 2° blaze angle and Littrow mounting (2K = 0). Solid curve, S-plane; dashed curve, P-plane.
For low blaze angle gratings (5° < qB < 10°), polarization effects will occur within their usable range (see Figure 9-6). In particular, a strong anomaly is seen near l/d = 2/3. Also observed is the theoretical S-plane theoretical efficiency peak of 100% exactly at the nominal blaze, combined with a P-plane peak that is lower and at a shorter wavelength. It is characteristic of all P-plane curves to decrease monotonically from their peak toward zero as l/d ® 2, beyond which diffraction is not possible (see Eq. (2-1)). Even though the wavelength band over which 50% efficiency is attained in unpolarized light is from 0.67lB to 1.8lB, gratings of this type (with 1200 groove per millimeter, for example) are widely used, because they most effectively cover the wavelength range between 200 and 800 nm (in which most ultraviolet-visible (UV-Vis) spectrophotometers operate).
Figure 9-6. Same as Figure 9-5, except 9° blaze angle.
A typical efficiency curve for a medium blaze angle grating (10° < qB < 18°) is shown in Figure 9-7. As a reminder that for unpolarized light the efficiency is simply the arithmetic average of the S- and P-plane efficiencies, such a curve is shown in this figure only, to keep the other presentations simple.
The low-anomaly blaze angle region (18° < qB < 22°) is a special one. Due to the fact that the strong anomaly that corresponds to the –1 and +2 orders passing off (l/d = 2/3) occurs just where these gratings have their peak efficiency, this anomaly ends up being severely suppressed (Figure 9-8). This property is quite well maintained over a large range of angular deviations (the angle between the incident and diffracted beams), namely up to 2K = 25°, but it depends on the grooves having an apex angle near 90°. The relatively low P-plane efficiency of this family of blazed gratings holds the 50% efficiency band from 0.7lB to 1.9lB.
Figure 9-7. Same as Figure 9-5, except 14° blaze angle. The curve for unpolarized light (marked U) is also shown; it lies exactly halfway between the S and P curves.
Figure 9-8. Same as Figure 9-5, except 19° blaze angle.
High blaze angle gratings (22° < qB < 38°) are widely used, despite the presence of a very strong anomaly in their efficiency curves (Figure 9-9). For unpolarized light, the effect of this anomaly is greatly attenuated by its coincidence with the P-plane peak. Another method for reducing anomalies for such gratings is to use them at angular deviations 2K above 45°, although this involves some sacrifice in efficiency and wavelength range. The 50% efficiency is theoretically attainable in the Littrow configuration from 0.6lB to 2lB, but in practice the long-wavelength end corresponds to such an extreme angle of diffraction that instrumental difficulties arise.
Figure 9-9. Same as Figure 9-5, except 26° 45' blaze angle.
Theoretically, all gratings have a second high-efficiency peak in the S-plane at angles corresponding to the complement of the blaze angle (90°–qB); in practice, this peak is fully developed only on steeper groove-angle gratings, and then only when the steep face of the groove is not too badly deformed by the lateral plastic flow inherent in the diamond tool burnishing process. The strong polarization observed at all high angles of diffraction limits the useable efficiency in unpolarized light, but it makes such gratings very useful for tuning lasers, especially molecular lasers. The groove spacing may be chosen so that the lasing band corresponds to either the first or second of the S-plane high-efficiency plateaus. The latter will give at least twice the dispersion (in fact the maximum possible), as it is proportional to the tangent of the angle of diffraction under the Littrow conditions typical of laser tuning.
Very-high blaze angle gratings (qB > 38°) are rarely used in the first order; their efficiency curves are interesting only because of the high P-plane values (Figure 9-10). In high orders they are often used in tuning dye lasers, where high
Figure 9-10. Same as Figure 9-5, except 46° blaze angle and 8° and 45° between the incident and diffracted beams (shown as light and heavy lines, respectively).
dispersion is important and where tuning through several orders can cover a wide spectral region with good efficiency. Efficiency curves for this family of gratings are shown for two configurations. With an angular deviation of 2K = 8°, the efficiency does not differ too much from Littrow; when 2K = 45°, the deep groove results in sharp reductions in efficiency. Some of the missing energy shows up in the zeroth order, but some of it can be absorbed by the grating.
A sinusoidal-groove grating can be obtained by the interferometric (holographic) recording techniques described in Chapter 4. Sinusoidal gratings have a somewhat different diffracted efficiency behavior than do triangular-groove gratings, and are treated separately.
It is convenient to consider five domains of sinusoidal-groove gratings,82 with progressively increasing modulation m, where
h is the groove height and d is the groove spacing:
Very low modulation gratings (m < 0.05) operate in the scalar domain,83 where the theoretical efficiency peak for sinusoidal grooves is only 34% (Figure 9 11). This figure may be readily scaled, and specification is a simple matter as soon as it becomes clear that the peak wavelength always occurs at lB = 3.4h = 3.4md. A blazed grating with an equivalent peak wavelength will require a groove depth 1.7 times greater.
Low modulation gratings (0.05 < m < 0.15) are quite useful in that they have a low but rather flat efficiency over the range 0.35 < l/d < 1.4 (Figure 9-12). This figure includes not only the infinite conductivity values shown on all previous ones, but includes the effects of finite conductivity by adding the curves for an 1800 g/mm aluminum surface. The most significant effect is in the behavior of the anomaly, which is the typical result of the finite conductivity of real metals.
Figure 9-11. First-order theoretical efficiency curve: sinusoidal grating, µ = 0.05 and Littrow mounting (2K = 0). Solid curve, S-plane; dashed curve, P-plane.
Figure 9-13 is a good example of a medium modulation grating (0.15 < m < 0.25). It demonstrates an important aspect of such sinusoidal gratings, namely that reasonable efficiency requirements confine first-order applications to values of l/d > 0.45, which makes them generally unsuitable for systems covering wide spectral ranges.
Figure 9-12. First-order theoretical efficiency curve: sinusoidal grating, aluminum coating, 1800 grooves per millimeter, µ = 0.14 and Littrow mounting. Solid curves, S-plane; dashed curves, P-plane. For reference, the curves for a perfectly conducting surface are shown as well (light curves).
Figure 9-13. Same as Figure 9-12, except µ = 0.22 and 8° between incident and diffracted beams (2K = 8°).
Over this restricted region, however, efficiencies are comparable to those of triangular-groove gratings, including the high degree of polarization. This figure also demonstrates how a departure from Littrow to an angular deviation of 2K = 8° splits the anomaly into two branches, corresponding to the new locations of the –1 and +2 order passing-off conditions.
High modulation gratings (0.25 < µ < 0.40), such as shown in Figure 9-14, have the maximum useful first-order efficiencies of sinusoidal-groove gratings. Provided they are restricted to the domain in which higher orders diffract (i.e., l/d > 0.65), their efficiencies are very similar to those of triangular-groove gratings having similar groove depths (i.e., 26° < qB < 35°).
Figure 9-14. Same as Figure 9-12, except µ = 0.36.
Very-high modulation gratings (µ > 0.40), in common with equivalent triangular-groove gratings, have little application in the first order due to their relatively low efficiencies except perhaps over narrow wavelengths ranges and for grazing incidence applications.
9.4. THE EFFECTS OF FINITE CONDUCTIVITY [top]
For metal-coated reflection gratings, the finite conductivity of the metal is of little importance for wavelengths of diffraction above 4 µm, but the complex nature of the dielectric constant and the index of refraction begin to effect efficiency behavior noticeably for wavelengths below 1 µm, and progressively more so as the wavelength decreases. In the P-plane, the effect is a simple reduction in efficiency, in direct proportion to the reflectance. In the S-plane, the effect is more complicated, especially for deeper grooves and shorter wavelengths.
Figure 9-15 shows the first-order efficiency curve for a widely-used grating: 1200 g/mm, triangular grooves, medium blaze angle (qB = 10°), coated with
Figure 9-15. First-order theoretical efficiency curve: triangular-groove grating, aluminum coating, 1200 grooves per millimeter, 10° blaze angle and 2K = 8°. Solid curves, S-plane; dashed curves, P-plane. For reference, the curves for a perfectly conducting surface are shown as well (light curves).
aluminum and used with an angular deviation of 8°. The finite conductivity of the metal causes a reduction in efficiency; also, severe modification of the anomaly is apparent. It is typical that the anomaly is broadened and shifted toward a longer wavelength compared with the infinite conductivity curve. Even for an angular deviation as small as 8°, the single anomaly in the figure is separated into a double anomaly.
For sinusoidal gratings, the situation is shown in Figures 9-12 and 9-14. Figure 9-13 is interesting in that it clearly shows a series of new anomalies that are traceable to the role of aluminum.
With scalar domain gratings (either qB < 5° or µ < 0.10), the role of finite conductivity is generally (but not always) to reduce the efficiency by the ratio of surface reflectance.84
9.5. DISTRIBUTION OF ENERGY BY DIFFRACTION ORDER [top]
Gratings are most often used in higher diffraction orders to extend the spectral range of a single grating to shorter wavelengths than can be covered in lower orders. For blazed gratings, the second-order peak will be at one-half the wavelength of the nominal first-order peak, the third-order peak at one-third, etc. Since the ratio l/d will be progressively smaller as |m| increases, polarization effects will become less significant; anomalies are usually negligible in diffraction orders for which |m| > 2. Figures 9-16 and 9-17 show the second- and third-order theoretical Littrow efficiencies, respectively, for a blazed grating with qB = 26°45'; they are plotted as a function of ml/d in order to demonstrate the proper angular ranges of use. These curves should be compared with Figure 9-9 for corresponding first-order behavior.
For gratings with sinusoidally shaped grooves, higher orders can also be used, but if efficiency is important, the choice is likely to be a finer pitch first-order grating instead. When groove modulations are very low (so that the grating is used in the scalar domain), the second-order efficiency curve looks similar to Figure 9-18, except that the theoretical peak value is about 23% (instead of 34%) and occurs at a wavelength 0.6 times that of the first-order peak, which corresponds to 2.05h (instead of 3.41h), where h is the groove depth. Successive higher-order curves for gratings with sinusoidal grooves are not only closer together, but drop off more sharply with order than for gratings
Figure 9-16. . Second-order theoretical efficiency curve: 26° 45' blaze angle and Littrow mounting. Solid curve, S-plane; dashed curve, P-plane.
Figure 9-17. Same as Figure 9-16, except third order.
Figure 9-18. Second-order theoretical efficiency curve: sinusoidal grating, µ = 0.36 and Littrow mounting. Solid curve, S-plane; dashed curve, P-plane.
Figure 9-19. Same as Figure 9-18, except third order.
with triangular grooves. For sufficiently deeply modulated sinusoidal grooves, the second order can often be used effectively, though (as Figure 9-18 shows) polarization effects are relatively strong. The corresponding third-order theoretical curve is shown in Figure 9-19.
9.6. USEFUL WAVELENGTH RANGE [top]
The laws governing diffracted efficiency are quite complicated, but a very rough rule of thumb can be used to estimate the useful range of wavelengths available on either side of the blaze (peak) wavelength lB for triangular-groove gratings.
For coarse gratings (for which d ³ 2l), the efficiency in the first diffraction order is roughly half its maximum (which is at lB) at 2lB/3 and 3lB/2. Curves of similar shape are obtained in the second and third orders, but the efficiencies are typically 20% less everywhere, as compared with the first order.
Grating of fine pitch (d » l) have a somewhat lower peak efficiency than do coarse gratings, though the useful wavelength range is greater.
9.7. BLAZING OF RULED TRANSMISSION GRATINGS [top]
Because they have no metallic overcoating, triangular-groove transmission gratings display far simpler efficiency characteristics than do their ruled counterparts. In particular, transmission gratings have efficiency curves almost completely free of polarization effects.
The peak wavelength generally occurs when the direction of refraction of the incident beam through a groove (thought of as a small prism) equals the direction dictated by the grating equation. [This is in direct analogy with the model of reflection grating blazing in that the grooves are thought of as tiny mirrors; see Section 2.8.] Due to the index of refraction of the grating, though, the groove angle exceeds the blaze angle for a transmission grating.
See Section 12.2 for more information on transmission gratings.
9.8. BLAZING OF HOLOGRAPHIC REFLECTION GRATINGS [top]
Although holographic gratings generally do not have the triangular groove profile found in ruled gratings, holographic gratings may still exhibit blazing characteristics (see, for example, Figure 9-18). For this reason it is not correct to say that all blazed gratings have triangular profiles, or that all blazed gratings are ruled gratings – blazing refers to high diffraction efficiency, regardless of the profile of the grooves or the method used to generate them.
This being said, there are some cases in which it would be preferable for a holographic grating to have a triangular groove profile rather than a sinusoidal profile. The method of using standing waves to record the grooves (see Section 4.2.1) was developed by Sheridon85 and improved by Hutley.86
Another useful technique for rendering sinusoidal groove profiles more nearly triangular is ion etching. By bombarding a surface with energetic ions, the material can be removed (etched) by an amount per unit time dependent on the angle between the beam and the local surface normal. The etching of a sinusoidal profile by an ion beam provides a continuously varying angle between the ion beam and the surface normal, which preferentially removes material at some parts of the profile while leaving other parts hardly etched. The surface evolves toward a triangular groove profile as the ions bombard it.87
Other method for generating blazed groove profiles have been developed,88 but the Sheridon method and the method of ion etching are those most commonly used for commercially-available gratings.
9.9. OVERCOATING OF REFLECTION GRATINGS [top]
The metallic coating on a reflection grating is evaporated onto the substrate. This produces a surface whose reflectivity is higher than that of the same metal electroplated onto the grating surface. The thickness of the metallic layer is chosen to enhance the diffraction efficiency throughout the spectral region of interest.
Most standard reflection gratings are furnished with an aluminum (Al) reflecting surface. While no other metal has more general application, there are a number of special situations where alternative surfaces or coatings are recommended. Gratings coated with gold (Au) and silver (Ag) have been used for some time for higher reflectivity in certain spectral regions, as have more exotic materials such as iridium (Ir), osmium (Os) and platinum (Pt).89
The reflectivity of aluminum drops rather sharply for wavelengths below 170 nm. While freshly deposited, fast-fired pure aluminum in high vacuum maintains its reflectivity to wavelengths shorter than 100 nm, the thin layer of oxide that grows on the aluminum (upon introduction of the coating to atmosphere) will cause a reduction in efficiency below about 250 nm.90 Fortunately, a method borrowed from mirror technology makes it possible to preserve the reflectivity of aluminum to shorter wavelengths.91 The process involves overcoating the grating with a thin layer of fast-fired aluminum, which is followed immediately by a coating of magnesium fluoride (MgF2) approximately 25 nm thick. The main purpose of the MgF2 coating is to protect the aluminum from oxidation. The advantage of this coating is especially marked in the region between 120 and 200 nm. While reflectivity drops off sharply below this region, it remains higher than that of gold and comparable to that of platinum, the most commonly used alternative materials, down to 70 nm.
Overcoating gratings so that their surfaces are coated with two layers of different metals sometimes leads to a change in diffraction efficiency over time. Hunter et al.92 have found the cause of this change to be intermetallic diffusion. For example, they measured a drastic decrease (over time) in efficiency at 122 nm for gratings coated in Au and then overcoated in Al + MgF2; this decrease was attributed to the formation of intermetallic compounds, primarily AuAl2 and Au2Al. Placing a suitable dielectric layer such as SiO between the two metallic layers prevents this diffusion.
As mentioned elsewhere, fingerprints are a danger to aluminized optics. It is possible to overcoat such optics, both gratings and mirrors, with dielectrics like MgF2, to prevent finger acids from attacking the aluminum. These MgF2 coatings cannot be baked, as is customary for glass optics, and therefore must not be cleaned with water. Spectrographic-grade organic solvents are the only recommended cleaning agents, and they should be used sparingly and with care.
Single-layer and multilayer dielectric overcoatings, which are so useful in enhancing plane mirror surfaces, are less generally applicable to diffraction gratings, since in certain circumstances these coatings lead to complex guided wave effects.93 For wavelengths below 30 nm, in which grazing angles of incidence and diffraction are common, multilayer coatings can enhance efficiency considerably.94
9.11. THE RECIPROCITY THEOREM [top]
A useful property of grating efficiency is that embodied in the reciprocity theorem,95 which states that (under certain conditions) reversing the direction of the beam diffracted by a grating will leave its diffraction efficiency unchanged. Stated another way, the reciprocity theorem says that the grating efficiency for wavelength l in order m is unchanged under the transformation a « b. This equivalence follows from the periodic nature of the grating, and is strictly true for perfectly-conducting gratings and lossless dielectric gratings illuminated by an incident plane wave.96
Three consequences of the reciprocity theorem should be noted:97
9.12. CONSERVATION OF ENERGY [top]
The principle of conservation of energy requires that all of the energy incident on a diffraction grating be accounted for; this can be represented mathematically (in terms of intensities), considering a single wavelength l, as
where the summation is over all diffraction orders m that propagate (i.e., for a reflection grating, all orders for which the diffraction angle b (m) satisfies the inequalities -90º £ b (m) £ +90º). Here the terms Iabsorbed and Iscattered are the "losses" due to absorption of energy by the grating and by scattering, respectively.
It is important to recognize that Eq. (9-2) holds only for the case in which the grating and the incident beam are fixed in space, and each diffraction order is diffracted through a unique diffraction angle given by Eq. (2-4) with constant a. Eq. (9-2) does not hold true when the intensity of each diffraction order m is measured in the Littrow configuration, since in this case the incidence angle a is changed for each order. Care must be taken when adding intensities (or efficiencies) in several orders for a single wavelength: the sum of these intensities is not conserved according to Eq. (9-2) unless the grating and incident beam remain fixed while a detector is moved (in angle) from one order to the next to take the intensity measurements.
Eq. (9-2) can be used to advantage by designing an optical system for which only two diffraction orders propagate: order m = 1 (or m = –1) and order m = 0 (which always exists: |a | < 90º implies |b (0)| < 90º, since b (0) = –a). This case requires a small groove spacing d and an incidence angle a such that the diffraction angle for m = ±2 and for the other first order pass beyond 90º: from Eq. (2-1) this requires
for m = 2, m = –2 and m = –1 (assuming that m = 1 is the order chosen to propagate). For such a system, Eq. (9-2) simplifies to become
where the term Idiff(0) corresponds to the reflected intensity and Idiff(1) corresponds to the intensity diffracted into the m = 1 order. Choosing a groove profile that reduces the reflected intensity Idiff(0) will thereby increase the diffracted intensity Idiff(1).
Eq. (9-2) is generally useful in measuring grating efficiency, but in the presence of anomalies (see below) they can lead to considerable inaccuracies.
9.13. GRATING ANOMALIES [top]
In 1902. R. W. Wood observed that the intensity of light diffracted by a grating generally changed slowly as the wavelength was varied, but occasionally a sharp change in intensity was observed at certain wavelengths.99 Called anomalies, these abrupt changes in the grating efficiency curve were later categorized into two groups: Rayleigh anomalies and resonance anomalies.100
9.13.1. Rayleigh anomalies
Lord Rayleigh predicted the spectral locations where a certain set of anomalies would be found: he suggested that these anomalies occur when light of a given wavelength l' and spectral order m' is diffracted at |b | = 90° from the grating normal (i.e., it becomes an evanescent wave, passing over the grating horizon). For wavelengths l < l', |b | < 90°, so propagation is possible in order m' (and all lower orders), but for l > l' no propagation is possible in order m' (but it is still possible in lower orders). Thus there is a discontinuity in the diffracted power vs. wavelength in order m' at wavelength l, and the power that would diffract into this order for l > l' is redistributed among the other propagating orders. This causes abrupt changes in the power diffracted into these other orders.
These Rayleigh anomalies, which arise from the abrupt redistribution of energy when a diffracted order changes from propagating (|b | < 90°) to evanescent (|b | > 90°), or vice versa, are also called threshold anomalies.101
9.13.2. Resonance anomalies
The second class of anomalies, which are usually much more noticeable than Rayleigh anomalies, are caused by resonance phenomena,102 the most well-known of which are surface excitation effects.103 At the interface between a dielectric and a metal, there are specific conditions under which a charge density oscillation ("electron wave") can be supported, which carries light intensity away from the incident beam and therefore decreases the diffraction efficiency of the grating. The efficiency curve would show a sharp drop in intensity at the corresponding conditions (see Figure 9-20).
Figure 9-20. . A typical (simplified) efficiency curve showing a sharp drop where the conditions are met for surface plasmon resonance. A narrow spectral region is shown; the efficiency curve would appear to increase monotonically if the resonance condition were not met.
For a resonance anomaly to exist, a resonance condition must be met – this places restrictions on the wavelengths (and incidence angles) that will exhibit resonance effects for a given groove profile and refractive indices. This results from the fact that in this phenomenon – the surface plasmon resonance (SPR) effect104 – the electromagnetic field that propagates along the metal-dielectric interface extends into each medium, so the characteristics of this propagating wave depend on the material conditions near the interface. This useful feature of SPR has led to its use in a number of sensing applications,105 such as biosensing106 and gas sensing.107 SPR can also be used to characterize the surface profile of the grating itself, especially by probing the diffraction effects due to higher harmonics in the periodic structure on the surface of the grating.108
While diffraction gratings generally do not convert incident P-polarized light to S-polarized light (or vice versa) upon diffraction, it has recently been observed that such polarization conversion can occur if the grating is not illuminated in the principal plane (i.e., e ¹ 0 in Eq. (2-3)).109 In this case, called conical diffraction (see Section 2.1), resonance effects can lead to a strong polarization conversion peak (e.g., a sharp trough in the S-polarized efficiency curve coincident with a sharp peak in the P-polarized efficiency curve).
9.14. GRATING EFFICIENCY CALCULATIONS [top]
Several techniques have been developed to calculate grating efficiencies, most of which have two characteristics in common: they employ Maxwell's equations whose boundary conditions are applied at the corrugated grating surface, and their difficulty in implementation varies in rough proportion to their accuracy. In this section only a brief mention of these techniques is provided - more details may be found in Petit110, Maystre111, and Loewen and Popov112.
Grating efficiency calculations start with a description of the physical situation: an electromagnetic wave is incident upon a corrugated surface, the periodicity of which allows for a multiplicity of diffracted waves (each in a different direction, corresponding to a unique diffraction order as described in Chapter 2). Efficiency calculations seek to determine the distribution of the incident energy into each of the diffraction orders.
Scalar theories of grating efficiency lead to accurate results in certain cases, such as when the wavelength is much smaller than the groove spacing (d << l ); the vectorial nature of optical radiation (manifest in the property of polarization) is not taken into account in this formalism.
Vector or electromagnetic theories can be grouped into two categories. Differential methods start from the differential form of Maxwell's equations for TE (P) and TM (S) polarization states, whereas integral methods start from the integral form of these equations. Each of these categories contains a number of methods, none of which is claimed to cover all circumstances.
Both differential and integral methods have been developed and studied extensively, and both have been implemented numerically and thoroughly tested against a wide variety of experimental data. Some of these numerical implementations are commercially available.
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