- Introduction to the calculation of structure factors S. In X-ray crystallography the structure factor F(hkl) of any X-ray reflection (diffracted beam) hkl is the quantity that expresses both the amplitude and the phase of that reflection. It plays a central role in the solution and refinement of crystal structures because it represents the quantity related to the intensity of the.
- ADVERTISEMENTS: The reliability of sampling data depends on the following factors. For this law to hold good, size of the sample should be large. If the size of the sample is small, conclusions.
- Powder diffraction data are usually presented as a diffractogram in which the diffracted intensity, I, is shown as a function either of the scattering angle 2θ or as a function of the scattering vector length q. The latter variable has the advantage that the diffractogram no longer depends on the value of the wavelength λ.
- Diffraction Depends On Which Factors
- What Factors Affect Diffraction
- Diffraction Depends On Which Factors Lead
- Diffraction Depends On Which Factors Make
It depends not only on the grating itself, but also substantially on operation conditions such as the optical wavelength and the angle of incidence. 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.
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Analyzing Beam Pathes Containing Gratings
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.
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.
Most common are reflection gratings, 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.
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.
Details of Diffraction at a Grating
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 ±2π / d, where d is the period of the grating structure.
An incident beam with an angle θ against the normal direction has a wave vector component k · sin θ along the plane of the grating, where k = 2π / λ and λ 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 · sin θ.Due to the grating's phase modulation, one can have additional reflected components with in-plane wave vector components −k · sin θ ± 2π / d.These correspond to the diffraction orders ±1.From this, one can derive the corresponding output beam angles against the normal direction:
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:
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.
The equations above may lead to values of sin θout 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.
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.
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.
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.
Distribution of the Output Power on the Diffraction Orders
Unlike a simple prism, a diffraction grating generally produces multiple output beams according to different diffraction orders.
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.
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.
The following section describes a common technique for optimizing diffraction efficiencies.
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.
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.
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.
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:
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.
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.
Fabrication Methods for Gratings
Gratings can be fabricated with different methods:
- A traditional technique is based on a ruling engine, a highly precise machine which mechanically imprints the required surface relief (a groove structure) on a metallic surface, for example, with a diamond tip.Although such ruled gratings are difficult to make with very small line spacings, they can be used for robust metallic blazed gratings with high diffraction efficiency and broad bandwidth.A disadvantage for use in grating spectrometers is that they cause substantial amounts of stray light due to surface irregularities.Also, it is difficult to ensure high uniformity over large surfaces.
- Laser micromachining can also be used for making relief gratings, although at somewhat larger dimensions – suitable mostly for long-wavelength applications.
- Holographic surface gratings are made with photolithographic techniques (or sometimes with electron-beam lithography), which allow finer grating structures.Simple holographic gratings have a sinusoidal phase variation and a low diffraction efficiency, but they cause only little stray light as their surfaces can be very regular.They can be made in a wide range of hard materials such as silica and various semiconductors, and advanced fabrication techniques can produce carefully controlled structures, such as blazed gratings.A high degree of uniformity over a large area is possible, but imperfections in the optics used for the fabrication process may produce superimposed “ghost gratings”.
- Holographic volume gratings have a periodic refractive index variation within a transparent medium.(See also the related article on volume Bragg gratings.)They can have high diffraction efficiency and low stray light, but can be sensitive to changes of temperature and humidity.Their sensitivity to humidity may be reduced by sealing them with suitable surface layers.
- It is also possible to replicate many gratings from a single master grating, which itself may be fabricated with a ruling engine or with a holographic technique.The replication process (usually involving some type of casting) can be much faster than the fabrication of the master, so that the method is well suited for mass production.
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.
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 .
Different Types of Gratings
Diffraction gratings can be distinguished from each other according to various aspects:
- There are reflection gratings, having a reflecting surface, and transmission gratings, where most of the incident light (diffracted and non-diffracted) is transmitted to the other side.
- Surface gratings have the grating structure on or near a surface, while volume gratings have it distributed in a larger volume.
- Also, one distinguishes surface relief gratings (utilizing a relief structure) from holographic gratings (with a refractive index variation).
- Different materials can be used.For example, there are gold gratings, where the reflecting layer consists of gold; other possible materials are e.g. aluminum, silver and speculum metal.Other gratings are based on purely dielectric structures.There are also hybrid metal–dielectric diffraction gratings, which can achieve a higher diffraction efficiency – particularly at shorter wavelengths, where the metal absorbs strongly.
- The label often reflects the used fabrication method – for example, there are ruled gratings, holographic gratings and replicated gratings.
- While in most cases the grating surface is plane (plane gratings), there are also gratings with a curved (e.g. spherical convex or concave) surface.This can be advantageous, for example, for achieving convenient imaging properties.There are also special aberration-corrected gratings.
- There are special gratings which are optimized for certain applications.For example, echelle gratings are made with a relatively low line density and used near gracing incidence and high diffraction orders.Grisms are prisms equipped with typically one surface grating.
- Some gratings, e.g. total internal reflection gratings, are based on special operation principles and named accordingly.
- Sometimes, gratings are labeled according to their applications.Examples are spectrometer gratings, beam combining gratings, laser tuning gratings and telecom gratings.
Important Properties of Diffracting Gratings
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.
Size and Uniformity, Wavefront Quality
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.
Diffraction Depends On Which Factors
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.
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.
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.
Spectral Resolution and Beam Radius
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 Δλ / λ is of the order of 1 / (m N) where m is the used diffraction order and N is the number of illuminated grating grooves.
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.
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.
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.
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.
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.
Thermal sensitivity can particularly be a problem in applications involving high-power laser radiation, such as spectral beam combining.
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.
Handling of Diffracting Gratings
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.
Applications of Diffraction Gratings
Diffraction gratings have many applications.In the following, some prominent examples are given:
Monochromators and Spectrometers
Many diffraction gratings are used in gratingmonochromators and spectrometers, where the wavelength-dependent diffraction angles are exploited.Figure 4 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.
Spectral separation can also be combined with imaging, as explained in the article on hyperspectral imaging.
Pairs of diffraction gratings can be used as dispersive elements without wavelength-dependent angular changes of the output.Figure 5 shows a Treacy compressor setup with four gratings, where all wavelength components are finally recombined ; 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.
Diffraction gratings are often used for wavelength tuning of lasers.For example, gratings in Littrow configuration can be used in external-cavity diode lasers.
Spectral Beam Combining
In spectral beam combining, one often uses a diffraction grating to combine radiation from various emitters at slightly different wavelengths into a single beam.
The RP Photonics Buyer's Guide contains 39 suppliers for diffraction gratings. Among them:
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.
Choose from our range of three diffraction gratings – the holographic research range, ruled research range and commercial range. Alongside our trio of ranges – available from stock or as custom-made optics – we also offer holographic diffraction grating films as customised components.
What Factors Affect Diffraction
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.
With a capacity to manufacture 500,000+ precision replicated optics, Optometrics, a Dynasil company, offers customers a reliable supply of diffractive, transmissive, and reflective optical components requiring high surface figure accuracy, nanometer range surface feature duplication, and low RMS surface roughness at a fraction of the original master surface production cost. Optometrics offers Echelle, filtered, holographic, infrared, ruled and transmission diffraction gratings.
NIL Technology can design, prototype and manufacture various diffraction gratings to customers' requirements. Examples include binary, blazed and slanted gratings with sub-wavelength period below 200 nm.
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|||T. Clausnitzer et al., “Highly efficient transmission gratings in fused silica for chirped-pulse amplification systems”, Appl. Opt. 42 (34), 6934 (2003), doi:10.1364/AO.42.006934|
|||M. Rumpel et al., “Linearly polarized, narrow-linewidth, and tunable Yb:YAG thin-disk laser”, Opt. Lett. 37 (20), 4188 (2012), doi:10.1364/OL.37.004188|
|||P. Poole et al., “Femtosecond laser damage threshold of pulse compression gratings for petawatt scale laser systems”, Opt. Express 21 (22), 26341 (2013), doi:10.1364/OE.21.026341|
|||M. Rumpel et al., “Broadband pulse compression gratings with measured 99.7% diffraction efficiency”, Opt. Lett. 39 (2), 323 (2014), doi:10.1364/OL.39.000323|
|||N. Bonod and J. Neauport, “Diffraction gratings: from principles to applications in high-intensity lasers”, Advances in Optics and Photonics 8 (1), 156 (2016), doi:10.1364/AOP.8.000156|
|||J. E. Harvey and R. N. Pfisterer, “Understanding diffraction grating behavior: including conical diffraction and Rayleigh anomalies from transmission gratings”, Optical Engineering 58 (8), 087105 (2019), doi:10.1117/1.OE.58.8.087105|
See also: diffraction, diffractive optics, transmission gratings, volume Bragg gratings, monochromators, spectrometers, chromatic dispersion, pulse compression, spectral beam combining
and other articles in the category general optics
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