An incident beam with an angle <$\theta$> against the normal direction has a wave vector component <$k \cdot \sin \theta$> along the plane of the grating, where <$k = 2\pi / \lambda$> and <$\lambda$> is the wavelength. Ordinary reflection (as would occur at a mirror) would lead to a reflected beam having the in-plane wave vector component <$-k \cdot \sin \theta$>. Due to the grating's phase modulation, one can have additional reflected components with in-plane wave vector components <$-k \cdot \sin \theta \pm 2\pi / d$>. These correspond to the diffraction orders ±1. From this, one can derive the corresponding output beam angles against the normal direction:

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Echelle gratings are a special type of echelette gratings (= blazed gratings), where the blaze angle is particularly large (beyond 45°). They are typically made with a relatively low groove density, used with a high angle of incidence, and for obtaining increased angular dispersion one utilizes high diffraction orders. They are mainly used in spectrometers and related types of instruments – often in combination with an ordinary grating for avoiding a confusion of light from multiple orders.

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Note that different sign conventions may be used for the diffraction order, so that there may be a minus sign in front of that term.

Generally, temperature changes result in changes of the line spacing, depending on the thermal expansion coefficients of the used materials. Different types of gratings can differ a lot in terms of thermal sensitivity.

Holographix works closely with our customers to assist them in the development of custom grating-based solutions from concept to production. We produce both reflection gratings and transmission gratings using our proprietary UV replication technology. Blazed, binary, sinusoidal and slanted grating profiles are available options.

Figure 3 shows how the number of diffraction orders depends on the ratio of wavelength and grating period, and on the angle of incidence. The number of orders increases for shorter wavelengths and larger grating periods.

It is instructive to consider the spatial frequencies of the position-dependent phase changes caused by a grating. In the simplest case of a sinusoidal phase variation, there are only two non-vanishing spatial frequency components with <$\pm2\pi / d$>, where <$d$> is the period of the grating structure.

If the damage threshold in terms of optical fluence is not as high as desired, one may operate a grating with a correspondingly larger beam area (or with nearly grazing incidence). That approach, however, also hits limitations, such as limited availability of large gratings or the required compactness of an apparatus.

For many applications, the diffraction efficiency is of high importance. This is the fraction of the incident optical power which is obtained in a certain diffraction order. It is often specified only for the desired diffraction order, not for the weaker unwanted orders. It depends not only on the grating itself, but also substantially on operation conditions such as the optical wavelength and the angle of incidence.

Note that Fresnel zone plates can also be regarded as a special kind of diffraction gratings, where one has circular structures instead of straight grating lines.

A diffraction grating is an optical device exploiting the phenomenon of diffraction, i.e., an kind of diffractive optics. It contains a periodic structure, which causes spatially varying optical amplitude and/or phase changes.

Headwall-manufactured OEM holographic diffraction gratings, spectrographs, and spectrometers maximize the quality of UV-VIS–NIR analysis. These master-quality diffraction gratings feature superior spectral and spatial resolution and extremely low scatter, meeting the demands of your high-performance instrument. We have more than 40 years of experience producing gratings and spectral engines for ourselves and for our OEM customers.

Diffraction gratings are often used for wavelength tuning of lasers. For example, gratings in Littrow configuration can be used in external-cavity diode lasers.

A promising approach is to avoid any materials with significant light absorption. For example, one can produce transmission gratings from purely dielectric materials with very low absorption and a high threshold for laser-induced damage.

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The handling of diffraction gratings – at least those with the grating near the surface – is usually relatively delicate. Grating surfaces are fairly sensitive e.g. against touching with hard objects or abrasive materials. It is thus also rather difficult to clean them; one should normally not try more than blowing off dust with clean, dry nitrogen or air. The deposition of any fat, oil or aerosol, for example, should be avoided wherever possible because it may be impossible to remove such deposits without damaging the grating.

The software RP Resonator also works with gratings. Find out e.g. the wavelength-dependent beam path in a single-pass configuration or within a laser resonator.

Thermal sensitivity can particularly be a problem in applications involving high-power laser radiation, such as spectral beam combining.

As the direction of each output beam – except for the zero-order beam – is wavelength-dependent, a diffraction grating can be used as a polychromator.

With the combination of ultra-pure fused silica substrates, all-dielectric coatings, and a unique grating design, Gitterwerk gratings offer a very high damage threshold with maximum diffraction efficiency (up to 99.5%). Beyond that, our unique mask-based full-field exposure process eliminates any stitching artifacts or period variations in our products, resulting in a perfect beam quality with excellent homogeneity. We cover the spectrum from 250 nm – 2500 nm with a series of line densities ranging from 600 l/mm up to 3000 l/mm, which both the optical performance and the mechanical dimensions can be tailored to the customer's specific requirements.

Figure 2 shows in an example case of a grating with 800 lines per millimeter, how the output angles vary with wavelength. For the zero-order output (pure reflection, <$m = 0$>), the angle is constant, whereas for the other orders it varies. The order <$m = 2$>, for example, is possible only for wavelengths below 560 nm.

Edmund Optics offers a wide range of transmission and reflection gratings for the ultraviolet (UV), visible and infrared (IR) regions. This includes both ruled and holographic gratings. We also offer polarization gratings, transmission grating beamsplitters and other variants.

A high diffraction efficiency for a particular diffraction order is essential for various applications. For example, a pulse compressor setup should not waste more of the generated pulse energy than is unavoidable. Also, high throughput of a spectrometer, enabled by using one or more highly efficient gratings, leads to a high detection sensitivity or possibly to reduced demands on the probe illumination, which is particularly important for battery-powered instruments.

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Grating 1 separates the input according to its optical frequencies (with passes for two different frequencies shown in the figure), and after grating 2 these components are parallel. Gratings 3 and 4 recombine the different components. The overall path length is frequency-dependent, and therefore the grating setup creates a substantial amount of anomalous chromatic dispersion, which may be used for dispersion compensation, for example.

This article treats mainly diffraction gratings where the diffraction occurs at or near the surface. Note that there are also volume Bragg gratings, where the diffraction occurs inside the bulk material.

Particularly for applications with pulsed lasers, it is important that gratings have a high enough optical damage threshold (see the article on laser-induced damage). Good power handling capabilities are tentatively in line with the requirement of low absorption losses, since only absorbed light has the potential for damaging a grating.

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The diffraction efficiency can depend on the line density and other factors, and there are various design trade-offs involving the diffraction efficiency and other properties.

Assuming an optimally blazed grating in Littrow configuration, the surfaces should be perpendicular to the incident beam. You can then use the fact that the optical path difference from groove to groove is <$\lambda/2$> (or an integer multiple thereof for higher-order operation), and geometrically calculate the groove depth from that.

Knight Optical has a range of stock diffraction gratings available, choose from our commercial, holographic and ruled research diffraction gratings for your application. If these do not meet your requirements, Knight Optical also offers custom diffraction gratings, enquire for further information.

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The Littrow configuration is used, for example, when a grating acts as an end mirror of a linear laser resonator. A given grating orientation fixes a wavelength within the gain bandwidth of the laser medium for which the resonator beam path is closed, i.e., laser operation is possible. This technique is used for making wavelength-tunable lasers – for example, external-cavity diode lasers.

Another possibility is to make a grating on top of a dielectric mirror structure, resulting in a reflecting grating mirror with very high diffraction efficiency [12].

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Generally, the diffraction efficiency is for the different orders can be polarization-dependent. This is particularly the case for reflection gratings, while transmission gratings often exhibit only a weak polarization dependence.

In a grating spectrometer, for example, one exploits the wavelength-dependent beam directions after a diffraction grating. The achievable wavelength resolution depends not only on the obtained angular dispersion (e.g. in units of microradians per nanometer), but also on the natural beam divergence angle: the smaller the divergence, the more precisely one can determine a change of angle. Therefore, a high wavelength resolution requires a large illuminated spot on the grating. One can show that the relative wavelength resolution <$\Delta \lambda / \lambda$> is of the order of <$1 / (m \: N)$> where <$m$> is the used diffraction order and <$N$> is the number of illuminated grating grooves.

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An important question is how the output power is distributed over the different diffraction orders. In other words, the diffraction efficiency for certain diffraction orders is of interest. This depends on the shape of the wavelength-dependent phase changes, and thus on the detailed properties of the grating grooves. In general, diffraction efficiencies can be calculated with diffraction theory.

Diffraction gratings can be optimized such that most of the power goes into a certain diffraction order, leading to a high diffraction efficiency for that order. For ruled gratings (see below), this optimization leads to so-called blazed gratings (echelette gratings), where the position-dependent phase change is described by a sawtooth-like function (with linear increases followed by sudden steps). The slope of the corresponding surface profile must be optimized for the given conditions in terms of input angle and wavelength. That optimization, however, can only work for a limited wavelength range.

Unlike a simple prism, a diffraction grating generally produces multiple output beams according to different diffraction orders.

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It is also possible to fabricate blazed holographic gratings, exhibiting a similar optimization of diffraction efficiency, although of course not related to a geometrical shape of the grooves.

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As explained above, the line density determines the angular positions (and even the existence) of the various diffraction orders. It may be limited by the used fabrication method, but can also be involved in design trade-offs.

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Most common are reflection gratings (or grating reflectors), where a reflecting surface has a periodic surface relief leading to position-dependent phase changes. However, there are also transmission gratings, where transmitted light obtains position-dependent phase changes, which may also result from a surface relief, or alternatively from a holographic (interferometric) pattern.

The phase difference of light reflected from two different grating lines is <$m \cdot 2\pi$> for the output beam direction. Essentially, this means no phase difference, as <$m$> is an integer number.

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As explained above, particularly high diffraction efficiencies are achieved with blazed gratings. Some transmission gratings also achieve very high diffraction efficiencies – sometimes even higher ones than for reflective gratings, essentially by avoiding absorption in metals.

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The alignment of diffraction gratings is often highly sensitive, requiring precise fine mechanics and high mechanical stability. The alignment sensitivity does not only depend on the grating itself (e.g. its line density), but also on various operation conditions and the application. The minimization of alignment sensitivity is often an important aspect for the design of optical arrangements involving gratings.

In the so-called Littrow configuration of a reflection grating, the diffracted beam – most often the first-order beam – is going back along the incident beam. This implies the following condition:

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If the grating's phase effect does not have a sinusoidal shape, one may have multiple diffraction orders <$m$>, and the output angles can be calculated from the following more general equation:

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Some diffraction gratings are specifically optimized for operation at or near the Littrow condition: they are blazed gratings (see above) for achieving a maximum diffraction efficiency. The shape of the grating grooves (assuming a ruled grating) is such that the linear parts of the structure are parallel to the wavefronts of the incident light. This also leads to a weak polarization dependence. Of course, that optimization can work only for a limited wavelength range, since the diffraction angles for other wavelengths will deviate from Littrow condition.

Many diffraction gratings are used in grating monochromators and spectrometers, where the wavelength-dependent diffraction angles are exploited. Figure 5 shows a typical setup of a monochromator. Artifacts in the obtained spectra can arise from confusion of multiple diffraction orders, particularly if wide wavelength ranges are recorded.

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In spectral beam combining, one often uses a diffraction grating to combine radiation from various emitters at slightly different wavelengths into a single beam.

It is also possible to fabricate a diffraction grating on a prism; the combination of a prism and a grating is sometimes called a “grism”. One may choose the parameters such that light at a certain center wavelength gets through the grism without any deflection.

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Pairs of diffraction gratings can be used as dispersive elements without wavelength-dependent angular changes of the output. Figure 6 shows a Treacy compressor setup with four gratings, where all wavelength components are finally recombined [2]; it can be used for dispersive pulse compression, for example. The same function is achieved with a grating pair when the light is reflected back with a flat mirror. (Note that such a mirror may be slightly tilted such that the reflected light is slightly offset in the vertical direction and can be easily separated from the incident light.) Such grating setups are used as dispersive pulse stretchers and compressors, e.g. in the context of chirped-pulse amplification. They can produce much larger amounts of chromatic dispersion than prism pairs, for example.

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The equations above may lead to values of <$\sin \theta_\rm{out}$> with a modulus larger than 1; in that case, the corresponding diffraction order is not possible. Figure 1 shows an example, where the diffraction orders −1 to +3 are possible.

Most used diffraction gratings only have dimensions of millimeters or a few centimeters, but it is also possible to fabricate very large gratings with dimensions of tens of centimeters or even more than one meter. A technical challenge is then to achieve a high uniformity over the whole grating area. Height uniformity is crucial for obtaining a high wavefront quality of diffracted beams.