定义：
包含周期性结构使光束发生衍射的光学元件。

衍射光栅是对光进行衍射的光学装置。它包含了一个周期性结构，引起空间振幅或者相位变化。常见的是反射光栅，其中反射表面具有周期性结构，产生的相位变化与位置有关。还存在透射光栅，这时透射光栅的相位变化与位置有关，也是由于存在表面的周期结构。 

图1：高功率超连续光源出射的白光，通过衍射光射后在空间上分散开从而可以显示其光谱含量。采用烟雾机可以看到光束路径。 

目录
 

1. 衍射光栅细节描述
2. Littrow结构
3. 光谱分辨率和光束半径
4. 输出功率在各级衍射光束中的分布
5. 光栅的制备方法
6. 衍射光栅的应用

衍射光栅细节描述 
有时需要考虑光栅引起的与位置有关的相位变化的空间频率。对于最简单的正弦相位变化，只存在两个空间频率部分为±2π / d，其中d是光栅结构的周期。 
入射角为θ的光在光栅平面的波矢分量为k • sin θ，其中 k = 2π / λ ， λ 是光波长。正常的反射会得到反射光的波矢分量为−k • sin θ。
由于光栅的相位调制，反射光还包含另一个平面波矢分量为−k • sin θ ± 2π / d的部分。它对应的是衍射级为±1。因此，可以得到出射光与法线之间的夹角满足： 

图2：光栅各级衍射光束。

如果光栅的相位变化不是正弦形状，这时存在多级衍射，出射角可以由下面这个更一般的公式来计算： 

对于衍射级可以采用不同的符号规则，因此有些项的前面可能会有负号。 
从上面方程可能会得到sin θout 值大于1，这时对应的衍射级是不存在的。图2给出了一个例子，只有衍射级-1~+3是存在的。 

图3：反射光束的输出角度与波长的关系。入射光束具有固定的入射角为 25°。 
图3中的光栅周期为每毫米800条，曲线描述的是出射角随波长的变化。对于零级衍射（纯反射，m=0），角度是常数，而其它阶的角度则随着波长变化而发生变化。例如二阶衍射m=2，只有当波长小于560 nm时才会发生。 
图4表明衍射级的数目与波长和衍射光栅周期比值的关系，以及与入射角的关系。波长越短，光栅周期越大，对应的衍射级的数目越大。 

图4：非零级衍射光栅的彩色编码数与波长除以光栅周期的关系。 

Littrow结构 
在Littrow结构的反射光栅中，衍射光栅（通常为一级光束）沿着入射光束方向返回。因此满足条件： 
这种结构通常用于，例如作为激光器谐振腔的端反射镜。一个给定的光栅方向可以确定激光介质的增益带宽内的某一波长，在该波长处谐振腔光束路径是闭合的，也就是说，可以实现激光产生。这项技术可用于波长调谐激光器中，例如，外腔二极管激光器。 

光谱分辨率和光束半径 
在光栅光谱仪中，利用了从衍射光栅得到的光束方向与波长有关这一性质。这时波长分辨率不仅与角色散（单位为微弧度每纳米）有关，还依赖于光束发散角：发散角越小，能越精确的得到角度变化。
因此，高的波长分辨率需要在光栅上的照明光斑很大。相对波长分辨率Δλ / λ处于的 1 / (m N)量级，其中m为衍射级数，N是照射的光栅刻槽数目。 

输出功率在各级衍射光束中的分布 
弄清楚输出功率在各级衍射光束中的分布是非常重要的。换句话说，需要确定某一级光束的衍射效率。这依赖于与波长有关的相位变化的形状。通常情况下，可以采用不同的衍射理论来计算衍射效率。 
可以优化衍射光栅使几乎所有的光功率都分布在某一级衍射光束上，因此在该级的衍射效率很高。
这称为闪耀光栅（小阶梯光栅），这时相位变化可以采用锯齿波函数表示。在给定的入射角度和波长条件下，需要调节光栅表面的斜率来优化光栅。在上面所述的Littrow结构中，结构的线性部分与入射光的波前是平行的。 

光栅的制备方法 
可以采用以下方法来制备光栅： 

• 传统的方法是采用刻线机，在金属表面刻划所需的表面凹凸结构（凹槽结构）。尽管刻划光栅很难实现很小的间隔，它们可以作为具有很高堰塞湖效率坚固的金属闪耀光栅。它们用于光栅光谱仪中时显著的缺点就是会由于表面不规则而产生一些杂散光。 
• 全息表面光栅是采用光刻技术制备的，可以实现更精细的光栅结构。简单的全息光栅相位变化是正弦型的，因此衍射效率较低，但是由于它们表面非常规则因此基本不会产生杂散光。
• 它们可以由很多坚硬的材料制备而成，例如二氧化硅和半导体材料，并且先进的制备技术可以得到精确控制的结构，例如闪耀光栅。 
• 体积全息光栅是在透明介质中具有周期性的折射率结构。它们具有很高的衍射效率并且产生很少的杂散光，但是对温度和湿度变化非常敏感。将其在表面用合适的涂层包裹起来可以减小湿度的影响。 

还可以在棱镜上制备衍射光栅，棱镜与光栅结合起来的结构称为棱栅（grism）。可以选择合适的参数，例如具有特定波长的光，使其无偏斜的透过棱栅。 
另外还可以在介质反射镜表面制备光栅结构，得到具有很高反射率的反射光栅。 

衍射光栅的应用 
衍射光栅具有很多种应用。下面给出一些主要的例子： 

1. 用作光栅光谱仪，利用了衍射角度与波长有关的性质。图5给出了典型的装置图。得到的光谱中包含多级衍射，尤其是记录的波长范围很大时。

图5：Czerny-Turner单色仪的设计图。

1. 一对衍射光栅也可用作色散元件，其中输出的角变化与波长无关。图6中的示意图包含了四个光栅，所有的波长组分最后汇合一起。
2. 采用一对光栅，并且光由平面反射镜反射回来也可以得到与上相同的结果。（注意，这时反射镜可能需要略微倾斜，这样反射光会相对垂直方向有偏移，可以与入射光分离开。）这一光栅装置可用作色散脉冲延伸器或者压缩器，在啁啾脉冲放大器中会用到。相比与棱镜对，它们可以产生更大的色散。

图6：四个光栅也就是两个光栅对的装置图。光栅1根据波长分离入射光束（图中表示了入射两个波长不同的光束的路径），经过光栅2后两光束平行。光栅3和4将光束重新合在一起。总的路径长度与波长有关，因此这一光栅装置产生很大的色散。

1. 如上所述，衍射光栅（Littrow结构）可以用于激光器的波长调谐。
2. 在频谱组束中，可以采用衍射光栅将不同发射器发射的波长略有不同的光合成为一个光束。

Definition: optical components containing a periodic structure which diffracts light

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 (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.

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:

Figure 1: Output beams of all possible diffraction orders at a diffraction grating.

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 θ_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.

Figure 2: Output angles of a reflective diffraction grating with 800 lines per millimeter as functions of the wavelength. The incident beam has fixed angle of 25° against the normal direction.

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: Color-coded number of non-zero diffraction orders as a function of the wavelength divided by the grating period.

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.

Blazed Gratings

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.

Figure 4: The white-light output of a high-power supercontinuum source is spatially dispersed by a diffraction grating in order to demonstrate the spectral content. The beam path has been made visible with a fog machine. Source: NKT Photonics.

Echelle Gratings

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.

Littrow Configuration

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 [12].

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).
• Echelle gratings are made with a relatively low line density and used near gracing incidence and high diffraction orders.
• 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.
• Gratings may also be called according to the application for which they are designed. Some examples:
• Spectrometer gratings are for use in spectrometers, optimized for spectral resolution and suppression of unwanted diffraction orders.
• Compressor gratings are for linear pulse compression and may be optimized for a high damage threshold.
• Beam combining gratings are for coherent beam combining, optimized e.g. for high damage threshold.
• Laser tuning gratings are for use in tunable lasers, optimized e.g. for high diffraction efficiency and high damage threshold.
• Some telecom gratings are designed for minimum polarization dependence.
• Some gratings, e.g. total internal reflection gratings, are based on special operation principles and named accordingly.

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.

Grisms are prisms equipped with typically one surface grating.

Important Properties of Diffracting Gratings

Line Density

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 Efficiency

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.

Polarization Dependence

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.

Damage Threshold

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.

Thermal Properties

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.

Alignment Sensitivity

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 grating monochromators 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.

Figure 5: Design of a Czerny–Turner monochromator. More details are given in the article on monochromators.

Spectral separation can also be combined with imaging, as explained in the article on hyperspectral imaging.

Pulse Compression

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 [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.

Figure 6: A four-grating setup, consisting of two grating pairs. Grating 1 separates the input according to wavelengths (with passes for two different wavelengths 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 wavelength-dependent, and therefore the grating setup creates a substantial amount of chromatic dispersion, which may be used for dispersion compensation, for example.

Wavelength Tuning

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.

Bibliography