# Diffraction gratings

Post-publication activity

Curator: Daniel Maystre

Born in the eighteenth century (Rittenhouse, 1786), the diffraction grating has been one of the most valuable instruments in the history of science and technology. At one extreme, it has enabled the study of celestial bodies. On the other hand, it has been a crucial tool in the study of atomic and molecular structures. These outstanding capabilities are the consequence of one basic property: it can disperse light, i.e. separate the frequencies contained in light radiation, thus allowing the measurement of relative intensities.

## The basic property : dispersion of light

### Grating description

Figure 1 represents a diffraction grating. The periodic profile $$\mathcal{P}$$ of period d along the x-axis separates air from a grating material which is generally a metal or a dielectric. The y-axis is perpendicular to the average profile plane and the z-axis is the axis of invariance of the structure. We suppose that the incident light can be described by a sum of monochromatic radiations of different frequencies. Each of these can in turn be described in a time-harmonic regime, which allows us to use the complex notation (with an $$\exp(-i\omega t)$$ time-dependence). The electromagnetic properties of the grating material (assumed to be non-magnetic) are represented by its complex refractive index $$\nu$$ which depends on the wavelength $$\lambda=2\pi c/\omega$$ in vacuum ($$c=1/\sqrt{\varepsilon_0\mu_0}$$ being the speed of light, with $$\varepsilon_0$$ and $$\mu_0$$ the permittivity and the permeability of vacuum). This complex index respectively includes the conductivity (for metals) and/or the losses (for lossy dielectrics). It becomes a real number for lossless dielectrics.

In the air region, the grating is illuminated by an incident plane wave, assumed to be s-polarized (electric field parallel to the z-axis) for simplicity, but the analysis is similar in the other case (p-polarization). With these conventions, the incident electric field $$\overrightarrow{E^i} =E^i\hat z$$ ($$\hat z$$ unit vector of the z-axis) is given by $\tag{1} E^i=\exp\bigl( ik_0 x\sin(\theta)-ik_0 y\cos(\theta)\bigr) ,$

with $$\theta$$ being the angle of incidence, from the y-axis to the incident direction, measured in the counterclockwise sense, and $$k_0 = 2\pi/\lambda$$ (we take an index equal to unity for air).

### Plane wave expansion of the field, grating formula

Let us now introduce, without demonstration (Petit, 1980), three intuitive properties of the total electric field. Like the incident electric field $$\overrightarrow{E^i}$$ , the total field $$\overrightarrow{E}$$ is:

• invariant along the z-axis,
• parallel to the z-axis, thus $$\overrightarrow{E}=E(x,y)\hat z$$ .
• pseudo-periodic , i.e. by definition $\tag{2} E(x+d,y)=E(x,y)\exp\bigl( ik_0 d\sin(\theta)\bigr) .$

Notice that in normal incidence ($$\theta=0$$ ), pseudo-periodicity becomes ordinary periodicity, which in that case is a straightforward property since both grating and incident wave are periodic.

Using these three properties and Maxwell's equations, let us show that the field above the grating is a sum of plane waves. With this aim, we use the first two Maxwell equations$\tag{3} \nabla\times\overrightarrow{E}=i\omega\mu_0\overrightarrow{H},\qquad \nabla\times\overrightarrow{H}=-i\omega\varepsilon\overrightarrow{E},$

with $$\overrightarrow{H}$$ being the magnetic field and $$\varepsilon$$ being the permittivity, equal to $$\varepsilon_0$$ in air region and to $$\varepsilon_0 \nu^2$$ in the grating material. In the following, equations placed at the left and right-hand sides of (3) will be called first and second Maxwell equations respectively. Combining the two Maxwell equations the electric field satisfies in the entire space a Helmholtz equation$\tag{4} \nabla^2E+k^2E=0,$

with$k=\begin{cases}k_0 &\text{in the air} \\k_0 \nu &\text{in the material} \end{cases}$

Moreover, it can be deduced from equation (2) that $$E(x,y)\exp\bigl( -ik_0 x\sin(\theta)\bigr)$$ has a period d along the x-axis and can thus be expanded in a Fourier series$\tag{5} E(x,y)\exp\bigl( -ik_0 x\sin(\theta)\bigr) =\sum_{n=-\infty}^{+\infty} {E_n (y)\exp(2i\pi nx/d)}.$

Multiplying both members of equation (5) by $$\exp\bigl(ik_0 x\sin(\theta)\bigr)$$ yields $\tag{6} E(x,y)=\sum_{n=-\infty}^{+\infty} {E_n (y)\exp(i\alpha_n x)},\qquad \alpha_n=k_0 \sin(\theta)+2\pi n/d.$

Introducing this expression of $$E(x,y)$$ in equation (4), we find $\tag{7} \sum_{n=-\infty}^{+\infty}{\bigl( \mathrm{d^2}E_n (y)/\mathrm{d}y^2+\beta_n^2 E_n (y)\bigr) \exp(i\alpha_n x ) }=0, \qquad \beta_n=\sqrt{(k_0^2-\alpha_n^2)}.$

Multiplying both members of the above equation by $$\exp\bigl (-ik_0 x\sin(\theta)\bigr)$$, we obtain a Fourier series in the left-hand side. It is vital to bear in mind that the nullity of this Fourier series entails the nullity of its coefficients, provided that the Fourier series vanishes for any value of x. Thus, equation (7) must be satisfied for any value of x. Since this equation is satisfied in the air region only, $$y$$ must be greater than the ordinate $$y_M$$ of the top of the grooves. In other words, we must restrict the property to the region located above the dotted blue line of figure 1. Finally, in this region$\tag{8} \text{if } y>y_M,\qquad E_n(y)=I_n\exp(-i\beta_n y) + R_n\exp(+i\beta_n y),$

and therefore, using equation (6), $$\tag{9} \text{if } y>y_M,\qquad E(x,y)=\sum_{n=-\infty}^{+\infty}{\bigl( I_n\exp(i\alpha_n x-i\beta_n y) + R_n\exp(i\alpha_n x+i\beta_n y)\bigr)}.$$

Let us remark that equation (7) does not assign to $$\beta_n$$ a unique value. However, equation (9) shows that its determination can be chosen arbitrarily since a sign change does not modify the value of the field, provided that $$I_n$$ and $$R_n$$ are permuted. The constants $$\beta_n=\sqrt{(k_0^2-\alpha_n^2)}$$ being either real or purely imaginary, we impose the real or imaginary part to be positive. Equation (9) shows that the field above the grooves can be represented by a plane wave expansion. The propagation constants of the plane waves along the x- and y-axes are respectively equal to $$\alpha_n$$ and $$\pm \beta_n$$. Some of these plane waves must be rejected since they do not obey the radiation condition (Petit, 1980). This rather intuitive condition states that the scattered field $$E^s$$ defined in air region by $$E^s=E-E^i$$ cannot include plane waves propagating downwards or exponentially increasing at infinity. Assuming that the incident electric field has a unit amplitude, this condition entails that $$I_n=\delta_{n,0}$$ with $$\delta_{n,0}$$ denoting the Kronecker symbol. Finally, equation (9) becomes$\tag{10} \text{if } y>y_M,\qquad E(x,y)=\exp(i\alpha_0 x-i\beta_0 y) +\sum_{n=-\infty}^{+\infty}{ R_n\exp(i\alpha_n x+i\beta_n y)},$

the sum being the expression of the scattered field in air region. The unknown complex coefficients $$R_n$$ are the amplitudes of the reflected waves.

### Dispersion of light

The conclusion of the last section is that above the grooves, the field scattered by the grating takes the form of a sum of plane waves, each of them being characterized by its order n. According to equation (7), almost all these waves (an infinite number) are evanescent and propagate along the x-axis at the vicinity of the grating profile. They correspond to the orders n such that $$\alpha_n^2 \ge k_0^2$$, thus rendering $$\beta_n=i\sqrt{(\alpha_n^2-k_0^2)}$$ a purely imaginary number. Only a finite number of them, called y-propagative orders, propagate towards $$y=+\infty$$ , with $$\alpha_n^2 \le k_0^2$$ , thus $$\beta_n=\sqrt{(k_0^2-\alpha_n^2)}$$ being real. Let us notice that among these orders, the $$0^{th}$$ order always exists. It propagates in the direction specularly reflected by the mean plane of the profile, whatever the wavelength may be. In contrast, the other y-propagative orders are dispersive. Indeed, their propagation constants along the x- and y-axes are equal to $$\alpha_n$$ and $$\beta_n$$, in such a way that the diffraction angle $$\theta_n$$ of one of these waves, measured clockwise from the y-axis, can be deduced from $$\alpha_n = k_0 \sin(\theta_n)$$. Using the expression of $$\alpha_n$$ given by equation (6), the angle of diffraction is given by $\tag{11} \sin(\theta_n) = \sin(\theta)+n\displaystyle\frac{2\pi }{k_0 d}= \sin(\theta)+n\displaystyle\frac {\lambda}{d} .$

This is the famous grating formula, often deduced from heuristic arguments of physical optics.

The field below the grooves has the same characteristics: as the field above the grooves, it can be represented by a sum of plane waves. However, it does not contain incident waves and, due to the radiation condition, the plane waves must propagate towards $$y=-\infty$$ or be attenuated in that direction. The wavenumber $$k_0$$ is replaced by $$k_0 \nu$$ , which yields$\tag{12} \text{if } y<0,\qquad E(x,y)=\sum_{n=-\infty}^{+\infty}{T_n\exp(i\alpha_n x-i\gamma_n y)} \qquad \gamma_n=\sqrt{(k_0^2 \nu^2 -\alpha_n^2)}.$

It is important to notice that this equation is obtained by assuming that the constant propagations $$\alpha_n$$ inside the material are the same as those in the air. This property is the direct consequence of pseudo-periodicity (equation (2)) which is valid in the entire space.

If the grating material is a lossless dielectric, the directions of propagation of the transmitted field obey a grating formula as well. This formula is similar to equation (11) , but since the constants of propagation $$\alpha_n$$ remain the same as in the air and since the wavenumber $$k_0$$ is multiplied by $$\nu$$, the angles of transmission $$\theta'_n$$ can be deduced from $$\alpha_n = k_0 \nu \sin(\theta'_n)$$, which yields$\tag{13} \nu \sin(\theta'_n) = \sin(\theta)+n\displaystyle\frac{2\pi }{k_0 d}= \sin(\theta)+n\displaystyle\frac {\lambda}{d} .$

When the grating material is metallic, the transmitted plane waves are absorbed by the metal and the y-propagating orders below the grooves no longer exist anymore.

In conclusion of this section, the reflected and transmitted fields include, outside the grooves, a finite number of plane waves propagating to infinity with scattering angles given by the grating formulae. All the orders are dispersive, except the $$0^{th}$$ orders. The reflected $$0^{th}$$ order takes the specular direction while for a lossless material, the transmitted $$0^{th}$$ order takes the direction transmitted by a plane interface. Consequently, a polychromatic incident plane wave generates in a given order a sum of plane waves scattered in different directions, i.e. a spectrum. The measurement of the intensity along this spectrum allows one to determine the spectral power of the incident wave, provided that the grating has been calibrated. The calibration, which can be achieved experimentally (using a known polychromatic incident plane wave) or theoretically (from the grating shape and the refractive index) provides the intensity ratio between the scattered wave in the spectral order and the incident wave, versus wavelength. This ratio is called efficiency of the grating in the given order and is smaller than unity. Of course, precise measurements of the power spectrum require large efficiencies inside the spectrum range. This requirement, which is crucial in spectroscopy, is not easy to obtain a priori. Indeed, we have to bear in mind that the efficiency of a dispersive order is always challenged by the $$0^{th}$$ order at least. This remark reveals the vital importance of the electromagnetic theory of gratings, which allows one to calculate the efficiency curve in a given order, and thus providing vital information about the spectroscopic quality of the grating before fabrication. Of course, this grating must be constructed using available technologies, a fact which points out that the progress of grating fabrication is vital.

## A fundamental instrument of science and technology

### Spectroscopy

For a long time, diffraction gratings have been considered as dispersive components of optics and used as tools for spectroscopic purposes only. The dispersive properties of gratings play a key role in many optical instruments, the most important being spectrometers and monochromators. Figure 2: The constant deviation mounting. The entrance and exit slits are fixed and the grating rotates around an axis perpendicular to the figure

In these instruments, the dispersion of light is not obtained using a fixed angle of incidence mounting as in figure 1. Indeed, for this configuration, the light source and the grating are fixed, therefore power measurements of the spectrum require a rotation of the detector. It is much easier to keep the source and the detector fixed and to rotate the grating, as illustrated in figure 2. The polychromatic light source is placed on the entrance slit, which is located on the focus of the lens L1. the grating is illuminated under the variable incidence $$\theta$$ by a parallel beam and generates spectra corresponding to the various propagative orders with $$n \neq 0$$ . The light which goes to the exit slit contains a set of wavelengths corresponding to different orders n, each of these wavelengths satisfying the grating formula (equation (11)) with angle of diffraction $$\theta_n=-D-\theta.$$ With our notation, D is the fixed angle between the axes of lenses $$L_1$$ and $$L_2$$, measured in the counterclockwise sense. It can be easily shown that, using the spectrum in the $$-1^{st}$$ order, a single wavelength is selected, provided that the range of wavelengths in the polychromatic light is not too large (Hutley, 1982). This wavelength is given by$\tag{14} \sin(\theta_{-1}) = \sin(\theta)-\displaystyle\frac {\lambda}{d} .$

Thus, the wavelength is linked to the angle of incidence by the equation $\tag{15} \lambda = d\bigl (\sin(\theta)+\sin(D+\theta) \bigr).$

In general, the deviation D is small and this constant deviation mount is identified as the Littrow mount, in which the deviation D vanishes. In this mounting, the incident wave and the $$-1^{st}$$ order have opposite directions and the wavelength is given from the incidence angle by $\tag{16} \lambda = 2d\sin(\theta).$

It can be verified from the grating formula that the $$0^{th}$$ and $$-1^{st}$$ orders are the only y-propagative orders in Littrow mount as long as $$\lambda>2d/3$$ (i.e. equivalently $$\theta>19.5°$$ ). This characteristic holds until the maximum wavelength $$\lambda=2d$$ (or equivalently $$\theta=90°$$ ). This range of wavelength is especially interesting for a spectroscopist using a metallic grating since the scattered power is shared between the $$-1^{st}$$ and $$0^{th}$$ orders only. As a consequence, the so-called blaze effect may be expected, at least for a few wavelengths in the spectrum.

The blaze effect is obtained when the $$-1^{st}$$ order is the only reflected order, the intensity in the $$0^{th}$$ order being null. A simple means to get the blaze effect is to use an echelette grating, as shown in figure 3 (Maréchal and Stroke, 1959). This kind of grating was for a long time the only manufactured grating. Its profile is made of two plane orthogonal facets, the large one making an angle b (blaze angle) with the mean plane of the grating. In Littrow mounting, when the angle of incidence is equal to the blaze angle, we can expect a very strong intensity in the $$-1^{st}$$ order since its direction of propagation is that of the specular reflection on the large facet. This heuristic blaze property can be demonstrated rigorously if the metal is considered as perfectly conducting (i.e. with an infinite conductivity) for p-polarization (magnetic field parallel to the grooves). In that case, there is no loss in the metal and thus, when the blaze effect arises, the totality of the incident power goes to the $$-1^{st}$$ order. It should be noticed that this blaze property holds, even when the number of reflected orders is greater than 2. For s-polarization (electric field parallel to the grooves), the efficiency is generally close (but not equal) to unity. It must be emphasized that the blaze effect can be obtained with other kinds of gratings, for example holographic gratings (with profiles close to a sinusoid) or lamellar gratings (with rectangular grooves), provided that the number of y-propagative orders is equal to 2 (Breidne and Maystre, 1980). When this is not the case, the echelette grating is, by far, the best one for blaze properties in Littrow mount.

In a spectrometer, the mount of figure 2 is used in order to know the spectral density of an unknown source, which can belong to a range extending from the ultraviolet to the infrared. For example, spectrographs are associated with terrestrial or embarked telescopes. They allow the astronomers to obtain crucial information on celestial objects like composition or speed. They also are of vital importance in the study of atomic or molecular structures via the light they emit. The Littrow mount is also used at the end of die laser cavities: a rotation of the grating permits the variation of the wavelength emitted by the laser (Hänsch, 1972).

X-ray spectroscopy is used in the study of synchrotron radiation. In that case, the period of the grating is much larger than the wavelength and, in general, a spectrum from large or very large orders must be used. Thus, in order to obtain significative efficiencies, it is necessary to employ mountings with specular reflection on the facet, i.e. mountings close to that of figure 3, with the direction of the employed order deduced from the incident direction by specular reflection on a facet, but with the incidence on the facet different from 0. Furthermore, due to the small reflectivity of metals under normal incidence in this domain, a large incidence angle on the large facet is used in general. Another means for enhancing the efficiency is to cover the large facet with alternate thin films of metallic and dielectric materials. More generally, the use of gratings with specular reflection on a facet are used in the so-called echelle gratings, viz. echelette gratings with large periods (Loewen and Popov, 1997). These gratings are often used in a large order, which increases the resolution. The resolution is the ability of a grating to separate two wavelengths, taking into account that a grating always has a finite size, thus that a monochromatic incident plane wave does not generate plane waves in the different orders but rather scattering lobes (Hutley, 1980).

### Non-spectroscopic applications

Among the non-spectroscopic applications of diffraction gratings, one of the oldest one is the use as a light polarizer in the infrared. A metallic grid composed of a periodic set of circular metallic rods is a very good reflector for s-polarized light above a cut-off wavelength, the sharpness of the cut-off increasing with rod thickness . One can have a heuristic explanation of this phenomenon by considering that the space located between two rods behaves like a truncated two-dimensional waveguide with variable section. Such a waveguide has a cut-off wavelength for s-polarization, above which it cannot transmit light, except by tunneling effect, which explains why the grid is a polarizer. The same high-pass filtering property holds for unpolarized light with inductive grids, i.e. biperiodic set of holes perforated in a metallic screen. This property may be used for example for solar absorption (McPhedran and Maystre, 1977). The inductive grid is placed above an absorbing body. Such a device can absorb solar energy if the cut-off wavelength is of the order of 1500 nm. Indeed, nearly all the solar power (mainly confined in the range 350-1500 nm) is transmitted to the absorbing body and, if this body is cooled by adequate liquid pipes, the thermic radiation it emits in the infrared above 1500 nm cannot escape from the absorber, thanks to the inductive grid.

A large part of the more recent applications of diffraction gratings is based on their ability to provoke strong resonance phenomena. One of the most striking examples of this property is the grating coupler, which can transmit the power of a light beam inside a dielectric waveguide (Neviere, Petit and Cadilhac, 1973).

The waveguide (figure 4) is formed by a dielectric film of index $$\nu_g$$(blue) placed above a dielectric substrate (orange) of index $$\nu_s < \nu_g$$ and covered by a very thin dielectric film (green) on a part of which has been made the grating profile. If the grating is removed, this device is a planar waveguide and it can guide light by successive total reflections on both sides of the central dielectric film (blue). There is no loss in the propagation, but consequently the power of an incident light beam cannot be transmitted inside the waveguide. Indeed, the propagation constant $$k_g$$ of a guided wave is greater than the wavenumber $$k_0$$ of the light in air region: as a consequence, the propagation constant along the x-axis of an incident plane wave or a beam, which it is smaller than $$k_0$$, cannot match $$k_g$$. We have seen that a grating can scatter evanescent waves having propagation constants $$\alpha_n$$ greater than $$k_0$$ (see equation (6)). Let us suppose that the entire upper profile of the waveguide is modulated. If, in this modulated waveguide, it exists an evanescent order n such that $$\alpha_n = k_g$$, the incident light can enter the waveguide through this evanescent order by resonant excitation. Conversely, the guided light can escape from the modulated waveguide and is thus exponentially attenuated in the propagation. However, if the modulated region has a finite width like in figure 4, the part of the incident power which is transmitted in the non-modulated region of the waveguide can propagate without any loss. This property is crucial in the technology of optical communications. Furthermore, due to the resonant process, the orders reflected and transmitted by the modulated waveguide present violent variations of efficiency and phase. Thus, if the planar waveguide is chosen in such a way that it transmits a large part of a polychromatic light, the modulated waveguide may present an efficiency very close to zero in the $$0^{th}$$ reflected order (like the planar waveguide), except at the vicinity of the resonant wavelength where it can reach unity (Popov et al., 1986). This phenomenon has been used for DWDM (Dense Wavelength Division Multiplexing) in optical communications (Boyko et al., 2009).

Another vital phenomenon of resonance, very close to that described in figure 4, is obtained by replacing the dielectric waveguide by a metal. Indeed, a plane interface between air and metal regions can guide a surface plasmon propagating along the interface with a propagation constant $$k_p > k_0$$ (Raether, 1988). In contrast with the case of the dielectric waveguide, the surface plasmon is attenuated during propagation, due to the losses inside the metal. If the surface is modulated in order to obtain a metallic grating, an incident beam may generate the surface plasmon by resonant excitation, provided that the propagation constant of an evanescent wave matches the propagation constant of the surface plasmon. As a consequence, the amplitude of the surface plasmon becomes very large and violent phenomena happen. First, a large part of the incident power may be absorbed by Joule effect. Secondly, the efficiencies vary very rapidly at the vicinity of the resonance when the wavelength or the incidence angle are modified. This phenomenon, classified as Wood’s anomaly, was discovered at the beginning of the 20th century (Wood, 1902). This is a major defect for spectroscopy but its selective nature constitutes the basis of Plasmonics. It has been shown theoretically and verified experimentally that the absorption can reach the totality of the incident power (Hutley and Maystre, 1976). Surprisingly, this total absorption is generally obtained with very shallow gratings.

Plasmonics has a crucial importance in nanophotonics, for example for its potential applications to optical communications. Indeed, the surface plasmon is nearly confined at the vicinity of the air-metal interface, thus it is possible to generate several uncoupled surface plasmons on closely separated regions of this air-metal interface. The propagation of surface plasmons in the visible, near infrared and near ultraviolet regions cannot be envisaged over long distances, due to the high losses, but distances of several dozen microns can be reached. This is sufficient for envisaging the use of surface plasmon propagation in photonic circuits (Ozbay, 2006).

Another kind of absorption can be achieved from very deep metallic gratings. Since the modulated region of the grating behaves like an impedance adaptor between air and metal regions, the grating acts like a non-selective light absorber. It can be used for example for increasing the efficiency of photovoltaic cells (Teperik et al., 2008). Finally, let us notice that excitation of surface plasmons by gratings is one of the keys that explain the phenomenon of extraordinary transmission by hole arrays like metallic inductive grids (Ebbesen et al., 1998).

It is not possible to give an exhaustive description of all the applications of diffraction gratings. Some of the most important have to be cited here, like their use as instruments for distance and shape measurements (Hutley, 1980), beam samplers for high power lasers, photolithography masks(see next section), light filters for recording color images as surface-relief structures (Knop, 1978), superlenses in nanophotonics, light pulse compression (Treacy, 1969). The diffraction grating is also the basic instrument of Diffractive Optics (Turunen and Wyrovsky, 1998).

## Rigorous electromagnetic theory

A rigorous grating theory must be deduced rigorously from the basic laws of electromagnetics in the form of a mathematical problem which can be solved on a computer. One may ask if a rigorous theory is necessary to investigate the properties of gratings? Many approximate theories are available. The most famous one uses the Kirchhoff approximation, which considers that any point of the grating profile behaves like an infinite plane interface tangent to the profile at that point (Beckmann and Spizzichino, 1987). These approximate theories are very simple to handle and to implement on computers. They may provide accurate results, under certain drastic conditions. Among these, the wavelength of light must be much smaller than the grating period, and the grating profile must be shallow. However, we have noticed in the preceding section that for spectroscopic purpose, gratings are generally used in conditions where the number of y-propagative orders is very small (often equal to 2). From the grating formula, this property entails that the wavelength of light has the same order of magnitude as the grating period and in that case such approximations fail.

### The RCWA method

First, we describe briefly one of the most popular rigorous theories of gratings: the Rigorous Coupled-Wave Analysis (RCWA), which is also the most simple one (Moharam and Gaylord, 1986). We have expressed the field above and below the intermediate region of the grating ($$0<y<y_M$$) as plane wave expansions in equations (10) and (12). If we can we express the field in the intermediate region, then the solution of the grating problem will be obtained by matching the expressions of the field in the 3 regions at ordinates $$y=0$$ and $$y=y_M$$. We start from Maxwell equations (see equation (3) ) and, for simplicity, we still suppose that the electric field is parallel to the z-axis. Bearing in mind that $$\nabla \times (E \hat z)=\nabla E \times \hat z$$ , we derive from the first Maxwell equation that $$\nabla E = i\omega \mu_0 \hat z \times \overrightarrow{H}$$ , which yields$\tag{17} \displaystyle \frac {\partial E}{\partial y} = i\omega\mu_0 H_x,\qquad \displaystyle \frac {\partial E}{\partial x} = -i\omega\mu_0 H_y,$

while the second Maxwell equation gives$\tag{18} \displaystyle \frac {\partial H_y}{\partial x} - \displaystyle \frac {\partial H_x}{\partial y} = -i\omega\varepsilon E, \qquad H_z=0.$

Finally, eliminating $$H_y$$ between equations (17) and (18) yields$\tag{19} \displaystyle \frac {\partial E}{\partial y} = i\omega\mu_0 H_x,\qquad \displaystyle \frac {\partial H_x}{\partial y} = \displaystyle \frac {i}{\omega \mu_0 } \displaystyle \frac {\partial^2 E}{\partial x^2}+i\omega \varepsilon E .$

In order to simplify this system of coupled partial derivative equations, we define $$\overset {\sim}{E} = E/(i\omega \mu_0)$$ and we deduce from equation (18)$\tag{20} \displaystyle \frac {\partial \overset {\sim}{E} }{\partial y} = H_x,\qquad \displaystyle \frac {\partial H_x}{\partial y} = - \displaystyle \frac {\partial^2 \overset {\sim}{E} }{\partial x^2}-k_0^2 \varepsilon_r \overset {\sim}{E} ,$

with $$\varepsilon_r$$ being the relative permittivity, equal to 1 in the air region and to $$\nu^2$$ in the grating material. Setting $$\overset {\sim}{E}(x,y) = \sum_{n=-\infty}^{+\infty}{ \overset {\sim}{E}_{n}(y) \exp(i\alpha_n x)},$$ and $$H_x (x,y) = \sum_{n=-\infty}^{+\infty}{ H_{x,n} \exp(i\alpha_n x)}$$, equation (20) yields$\tag{21} \displaystyle \frac { d\overset {\sim}{E}_n(y) } {dy} = H_{x,n}, \qquad \displaystyle \frac { d H_{x,n}} {d y} = - \alpha_n^2 \overset {\sim}{E}_n(y)-k_0^2 \sum_{m=-\infty}^{+\infty} \varepsilon_{r,n-m} \overset {\sim}{E}_m ,$

with $$\varepsilon_{r,p}$$ being the $$p^{th}$$ Fourier coefficient of $$\varepsilon_r$$, function of y. We obtain an infinite set of first order differential equations which can be expressed in the symbolic form $$d\mathbf{F}/dy=\mathbf{A}(y)\mathbf{F}$$ where $$\mathbf{F}$$ is an infinite column matrix containing successively the set of functions $$\overset {\sim}{E}_n$$ then the set of functions $$H_{x,n}$$ , and $$\mathbf{A}(y)$$ a matrix with elements deduced from equation (21).

In the case of lamellar gratings (figure 5a) , the Fourier coefficients $$\varepsilon_{r,n}$$ of the relative permittivity do not depend on y and thus the matrix $$\mathbf{A}$$ is constant in the intermediate region. As a consequence, the form of the fields in the intermediate region can be found easily from equation (21). This special case is very interesting since it strongly simplifies the numerical implementation of the method. Thus, our study will be restricted to this kind of grating in the following. A generalization of the method to other profiles can be realized, for example by approaching the profile by a set of thin lamellar gratings (figure 5b).

The system of coupled differential equations can be truncated to $$4N+2$$ order by limiting the integers n and m to the range (-N, +N), in such a way that $$\mathbf{A}$$ becomes a square matrix$\tag{22} d\mathbf{F}/dy=\mathbf{A}(y)\mathbf{F},\qquad \text{with}\qquad \mathbf{F}=\begin{bmatrix} \mathbf{e} \\ \mathbf{h}\end {bmatrix} ,$

where the column matrices $$\mathbf{e}$$ and $$\mathbf{h}$$ of size $$2N+1$$ contain the functions $$\overset {\sim}{E}_n$$ and $$H_{x,n}$$ respectively. The solutions of this set of differential equations form a vector space of dimension $$4N+2$$. Thus, any solution can be expressed in the form of a linear combination of $$4N+2$$ independent solutions of the system. It is straightforward to verify that these independent solutions $$\mathbf{S}_q$$ can be found from the eigenvectors $$\mathbf{V}_q$$ and eigenvalues $$\tau_q$$ of matrix $$\mathbf{A}$$ by setting $$\mathbf{S}_q=\mathbf{V}_q \exp(\tau_q y)$$ . Thus, using a computer library program to compute the eigenvectors and eigenvalues, any solution of the truncated system can be expressed in the form$\tag{23} \mathbf{F}=\sum_{m=1}^{4N+2} c_m \mathbf{S}_m .$

Equation (23) gives the form of the field in the intermediate region. In order to determine the unknown coefficients $$c_m$$ , we express the continuity of the fields at $$y=0$$ and $$y=y_M$$ . The expansions of the fields are given above and below the intermediate region by equations (10), (12), and the similar expressions of $$H_{x,n}$$ deduced from equations (10) and (12) using the first Maxwell equation. The field in the intermediate region is given by separating equation (23), into two sets of $$2N+1$$ equations in order to separate the column matrices $$\mathbf{e}$$ and $$\mathbf{h}$$ (see equation (22)). We obtain finally 4 sets of $$2N+1$$ equations$\tag{24} \forall n \in (-N,+N),\qquad \delta _{n,0}\exp(-i\beta_0 y_M) + R_n \exp(i\beta_n y_M) = \sum_{m=1}^{4N+2}{ c_{m}S_{N+1+n,m}(y_M)} ,$

$$\tag{25} \forall n \in (-N,+N),\qquad \displaystyle \frac {1}{\omega \mu_0}\Bigl(-\beta_0 \delta _{n,0}\exp(-i\beta_0 y_M)+ \beta_n R_n \exp(i\beta_n y_M)\Bigr) = \sum_{m=1}^{4N+2}{ c_{m}S_{3N+2+n,m}(y_M)} ,$$

$$\tag{26} \forall n \in (-N,+N),\qquad T_n = \sum_{m=1}^{4N+2}{ c_{m}S_{N+1+n,m}(0)} ,$$

$$\tag{27} \forall n \in (-N,+N),\qquad \displaystyle \frac {-1}{\omega \mu_0}\gamma_n T_n = \sum_{m=1}^{4N+2}{ c_{m}S_{3N+2+n,m}(0)} .$$

Eliminating the coefficients $$R_n$$ between equations (24) and (25) provides a first set of $$2N+1$$ equations with $$4N+2$$ unknown coefficients $$c_{m}$$ . The second set of $$2N+1$$ equations is obtained by eliminating the coefficients $$T_n$$ between equations (26) and (27). Finally, the linear system of $$4N+2$$ equations with $$4N+2$$ unknowns is solved using a computer library program. The amplitudes $$R_n$$ and $$T_n$$ of the reflected and transmitted orders deduce from coefficients $$c_{m}$$ using equations (24) and (26). Calculating the flux of the Poynting vector through a rectangle of length d along the x-axis, with a top in air region and a bottom in the grating material, the energy balance shows that the efficiency in the $$n^{th}$$ reflected order is equal to $$(\beta_n/\beta_0)\left|R_n\right|^2$$ and, when the grating material is a lossless dielectric, the efficiency of the $$n^{th}$$ transmitted order is equal to $$(\gamma_n/\beta_0)\left|T_n\right|^2$$ (Petit, 1980).

The RCWA is one of the most popular methods of grating theory, even though it may present some problems of stability in the visible, infrared and terahertz regions when the grating material is metallic: the amplitudes of the various orders do not converge as the parameter $$N$$ is increased. However, it should be noticed that considerable progress have been obtained to increase the stability (Li, 1996; Lalanne and Morris, 1996; Granet and Guizal, 1996; Popov and Nevière, 2003).

Before the RCWA, a very closely related method has been published, the differential method (Hutley et al., 1975). In fact this method remains quite similar to the RCWA up to equation (21). Then it solves the system of differential equations with coefficients depending on y by using classical algorithms.

### The integral method

Historically, the first rigorous method was achieved in the 60s. It is the integral method, which reduces the grating problem to an integral equation or a set of two coupled integral equations (Maystre, 1984; DeSanto, 1981). The main advantage of this method is that it can solve almost any grating problem, regardless of the grating material, the range of wavelength (from X-rays to microwaves) or the shape of the grating. Let us give an intuitive presentation of this method on the simple case of a s-polarized incident plane wave illuminating a perfectly conducting grating.

From a heuristic point of view, this method is based on the interpretation of scattering through two phenomena:

• The incident wave which illuminates the grating profile cannot penetrate the grating material and thus it generates on the grating profile a surface current density $$\overrightarrow{j}(\mathrm M') =j(\mathrm M')\hat z$$ which depends on the point $$\mathrm M'$$ of the profile.
• The surface current density $$j(M')$$ creates at any point $$P$$ of space a scattered field $$E^s(\mathrm P)\hat z$$.

The scattered field can be deduced from the current density on the profile, which generated it. Indeed, the grating profile can easily be represented by a superposition of short arc elements of length $$\mathrm dl'$$ centered around points of the profile. In figure 6, we show one of these elementary arcs (red line), centered around a point $$\mathrm M'$$. The field scattered by this arc at any point $$\mathrm P$$ of space can be approximated by the field which would be generated by a line current $$j(\mathrm M')\mathrm dl'$$ concentrated on a line parallel to the z-axis and including $$\mathrm M'$$, at least if the distance $$\mathrm {PM'}$$ is large with respect to $$\mathrm dl'$$. This approximation tends to a rigorous representation, including at the vicinity of the profile, as the arc length $$\mathrm dl'$$ tends to zero. The calculation in closed form of the field scattered at any point $$\mathrm P$$ of space by a unit current placed at point $$\mathrm M'$$ does not present any difficulty. From a mathematical point of view, it is the so called Green's function of the Helmholtz equation in cylindrical coordinates (namely a Hankel function), which depends on the distance $$\mathrm {PM'}$$ only. Let us denote this field by $$G(\mathrm {PM'})\hat z$$. By adding the contributions of all the elementary arcs of the profile, and assuming that each elementary arc length $$\mathrm dl'$$ tends to zero, it turns out that the field scattered at any point $$\mathrm P$$ of space can be written in the following integral form$\tag{28} E^s(\mathrm P) = \int\limits_{grating\;profile} G(\mathrm {PM'})j(\mathrm M')\mathrm dl' .$

Since the surface current density presents the same pseudo-periodicity as the field, it is straightforward to limit the integral on the infinite profile to a single period$\tag{29} E^s(\mathrm P) = \int\limits_{1\;period} K(\mathrm P,\mathrm M')j(\mathrm M')\mathrm dl' .$

where $$K(\mathrm P,\mathrm M')$$ depends now on $$\overrightarrow{\mathrm {PM'}}$$ and not on $$\mathrm {PM'}$$ only. $$K(\mathrm P,\mathrm M')$$ can be expressed as a sum of exponential functions (Petit, 1980; Maystre, 1984). Equation (29) allows one to show, after elementary calculations, that the scattered field above the grating profile can be represented as a plane wave expansion, which is nothing else than the sum in the right-hand member of (10). In addition, (29) provides the expression of the amplitudes $$R_n$$ of the plane waves (thus the efficiencies) from the surface current density $$j$$ through simple integrals.

The surface current density can be calculated from equation (29) as well, by writing that the total electric field at a point $$\mathrm {P}$$ in the air must tend to zero as $$\mathrm {P}$$ tends to a point $$\mathrm {M}$$ of the profile. Indeed, the tangential component of the electric field is continuous across the grating surface: since it vanishes on the side of the perfectly conducting material, it must also vanish on the air side of the interface. Here, the total electric field is parallel to the z-axis, thus it is tangential and vanishes$\tag{30} \int\limits_{1\;period} \lim_{\mathrm P \to \mathrm M} [K(\mathrm P,\mathrm M')] j(\mathrm M')\mathrm dl' + E^i(\mathrm M) = 0 .$

It turns out that, in that case, $$\lim_{\mathrm P \to \mathrm M} [K(\mathrm P,\mathrm M')]= K(\mathrm M,\mathrm M').$$ Equation (30) is an integral equation since the unknown function $$j(\mathrm M')$$ is placed inside the integral. The function $$K(\mathrm P,\mathrm M')$$ is called the kernel of the integral equation.

The number of available softwares based on this method is very small, a consequence of the difficulty to handle the theory, and above all the numerical implementation. The most drastic difficulty comes from the fact that the kernel is a series which is singular as $$\mathrm M$$ tends to $$\mathrm M'$$. Classical methods exist for solving the integral equation, the finite-elements method being the most widespread. The integral method, sometimes called theory of potentials or Green's functions method is the most popular method in electromagnetics.

### Some other methods

Some other methods are also wide-spread. A more recent theory, the C-method, has proved to be a simple, general and stable tool (Chandezon et al., 1982). It is based on a change of coordinates which reduces the grating profile to a coordinate axis and leads to a differential system of equations. A last popular method must be mentioned: the modal theories (Botten et al., 1981). In contrast with the other theories, it is restricted to special kinds of profiles. In practice, it has been developed for lamellar gratings. The theory takes advantage of the special profile to express the form of the fields inside the intermediate region in the form of a series with unknown coefficients. These coefficients are calculated through the inversion of a linear system of equations with coefficients in closed form. General methods of Electromagnetics have also been used to solve grating problems, like FDTD (Finite Difference Time Division) or finite elements method (Demézy et al, 2009).

Finally, a well known method must be cited: the Rayleigh method (Lord Rayleigh, 1907). This method, historically the first attempt at solving the grating problem rigorously, is based on the so-called Rayleigh hypothesis: the plane wave expansions above and below the intermediate region are valid in the entire regions above and below the grating profile. Thus, matching the expansions of the fields on the profile provide directly the amplitudes of the reflected and transmitted orders. It has been shown that the Rayleigh hypothesis may fail and in practice, the Rayleigh method leads to numerical instabilities, except for shallow gratings.

Let us notice that the expression “rigorous method” is not always well understood and sometimes criticized. Indeed, approximations are needed in the numerical implementation. For example, in the RCWA method, the expansions of the field are limited to $$2N+1$$ harmonics. Nevertheless, it must be emphasized that the theory is based on the elementary laws of Electromagnetics, without any approximation, in such a way that, by increasing $$N$$, numerical results can be obtained to an arbitrary level of accuracy, which is not the case for approximate methods in which theoretical assumptions or approximations are used.

## Manufacturing

### Echelette gratings

The first diffraction grating was made in 1786 (Rittenhouse, 1786). The American astronomer D. Rittenhouse saw a spectrum generated by hairs placed in the threads of two parallel screws. However, the production of high quality gratings was not achieved before the end of the 19th century (Rowland, 1882). H.A. Rowland used ruling engines to make metallic diffraction gratings having more than 100 000 grooves. Subsequently, the technology of ruled gratings, also called echelette gratings or blazed gratings (figure 3) accomplished much progress (the interested reader can refer to (Harrison, 1949)). For almost one century, this kind of grating remained in practice the only one to be produced and used for spectroscopy purpose . Echelette gratings are made using a diamond edge interferometrically controlled in order to draw parallel grooves periodically spaced on a thin layer of metal deposited on an optically flat glass substrate. More precisely, this technique is used to construct master gratings. Echelette gratings are made by replicating these master gratings via vacuum deposition on the master substrate of an intermediate and a metallic layers reproducing the master surface, then an epoxy-coated glass substrate. Finally, the replicated grating is separated from the master. Despite numerous improvements in the technology of ruling, making a master is not an easy task. A ruling engine is a very complex and sophisticated object of precision engineering and in most cases, it is a compromise between conflicting requirements. Moreover, the diamond edge can wear out during the long process of ruling (Harrison, 1949).

### Holographic gratings

Due to the fact that stellar spectroscopy with large telescopes requires the use of large high quality diffraction gratings, a new process for making gratings was initiated in Germany (Schmahl, 1974) and in France (Labeyrie and Flamand, 1969). The holographic (or interferometric) process uses a laser light illuminating a photoresist and, in contrast with ruled gratings, all the grooves are made simultaneously. The photoresist layer (figure 7) is exposed to the fringe system, which produces a variation of solubility. When it is developed in a suitable solvent, the removal of material depends on exposure, in such a way that the structure becomes a transmission dielectric grating. It can be converted into a reflection metallic grating by vacuum coating with a metal. The sensitivity of most photoresist materials strongly falls for wavelengths greater than 500 nm. Generally, holographic grating profiles are close to a sinusoid. However, the process is not completely linear and thus, the shape of the grooves is not perfectly sinusoidal, even though the variation of light intensity is perfectly sinusoidal, except for shallow gratings.

It is worth noticing that the finest period that can be obtained is $$\lambda/2$$, $$\lambda$$ being the wavelength of illumination of the photoresist. This spacing cannot be achieved in practice since it corresponds to an incidence of the beams equal to 90°, but a period of 0.6$$\lambda$$ is reached with an incidence of 60°. Figure 8: Top: surface of a 600 grooves/mm echelette grating (Jobin-Yvon) observed by electron-microscopy. Notice the big defect at center. Bottom: profile of a holographic grating (Jobin-Yvon)

Figure 8 shows the profile of a holographic grating (bottom) and the surface of a ruled grating (top). Holographic gratings are often preferred in monochromators because they lead to much less stray light. The stray light may be caused by random grating defects and results in unexpected wavelengths being detected by the spectrometer. On the other hand, both ruled and holographic gratings can provide adequate resolution powers (the resolution power is the ability of a grating to separate two close wavelengths and strongly depends on the total number of grooves).

### More recent technologies

Ruled and holographic gratings are basic components of gratings used in spectroscopy but more recent applications of gratings have required new fabrication processes. In general, these processes were developed originally for making microelectronic and nanoelectronic components like integrated circuits. A large part of these new tools belong to the frame of lithography. The technique of lithography is to reproduce a pattern printed on a mask on a suitable layer deposited on a substrate (Jaeger, 2002). With this goal, it uses particles to remove (directly or after using a chemical treatment) selective parts of this material after crossing the mask. A new material (for example a metal) may be deposited after the treatment. In photolithography, the particles are photons and the layer is a photoresist. For example, a photomask can be made of a fused silica blank covered with a chrome absorbing pattern. They are used in general below 400 nm. A better resolution (15 nm) is obtained using Extreme Ultraviolet Lithography (EUV) or X-ray Lithography. Other lithographic techniques use electrons (Electron Beam Lithography, Scanning Probe Lithography, Electron Beam Direct-Write Lithography…) or ions (Ion Beam Lithography).

Another technique, Reactive Ion-Etching (RIE), is used for grating fabrication (Oehrlein, 1986). A wafer platter is placed in a chamber. Electrically, the walls of the chamber are the ground and the wafer is isolated from them. A plasma is generated under low pressure by a radio frequency electromagnetic field striking a gas inside the chamber: the field ionizes the gas by removing electrons from molecules. Thus, the electrons go alternatively up and down in the chamber. When they strike the top of the chamber, they go to the ground potential. On the other hand, when they strike the isolated platter, they generate on it a negative charge, the plasma developing a positive charge, due to the higher concentration of positive ions. As a result, the ions move towards the wafer platter and they strike the surface. They can either provoke chemical reactions or sputter some material on the non-protected parts of the sample (for example, the protection can be made with photoresist). Since the motion of ions is nearly vertical, the process of etching acts vertically an can generate deep profiles, especially in the Deep Reactive Ion-Etching (DRIE).

Finally, let us mention a technology adapted to integrated optical circuits: Digital Planar Holography (DPH). Gratings are generated by computer on a waveguide using techniques of micro-lithography.