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Diffraction Gratings

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Definition: optical components containing a periodic structure which diffracts light

German: Beugungsgitter

Category: general optics

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A diffraction grating is an optical device exploiting the phenomenon of diffraction. 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.

This article treats those 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.

supercontinuum dispersed at a diffraction grating

Figure 1: 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. The photograph has been kindly provided by NKT Photonics.

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:

beams at a diffraction grating

Figure 2: Output beams of all possible diffraction orders at a grating.

output angle at 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:

output angle at diffraction grating

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 2 shows an example, where the diffraction orders −1 to +3 are possible.

diffraction angles vs. wavelength

Figure 3: 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 3 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.

number of diffraction orders of a diffraction grating

Figure 4: Color-coded number of non-zero diffraction orders as a function of the wavelength divided by the grating period.

Figure 4 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.

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:

Littrow 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.

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.

Distribution of the Output Power on the 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. 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. 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. In the Littrow configuration (see above), the shape is such that the linear parts of the structure are parallel to the wavefronts of the incident light.

Fabrication Methods for Gratings

Gratings can be fabricated with different methods:

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 reflectivity [2].

Applications of Diffraction Gratings

Diffraction gratings have many applications. In the following, some prominent examples are given:

Czerny-Turner monochromator

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

pairs of diffraction gratings

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.


[1]E. B. Treacy, “Optical pulse compression with diffraction gratings”, IEEE J. Quantum Electron. 5 (9), 454 (1969)
[2]M. Rumpel et al., “Linearly polarized, narrow-linewidth, and tunable Yb:YAG thin-disk laser”, Opt. Lett. 37 (20), 4188 (2012)
[3]P. Poole et al., “Femtosecond laser damage threshold of pulse compression gratings for petawatt scale laser systems”, Opt. Express 21 (22), 26341 (2013)
[4]M. Rumpel et al., “Broadband pulse compression gratings with measured 99.7% diffraction efficiency”, Opt. Lett. 39 (2), 323 (2014)
[5]N. Bonod and J. Neauport, “Diffraction gratings: from principles to applications in high-intensity lasers”, Advances in Optics and Photonics 8 (1), 156 (2016)

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See also: volume Bragg gratings, spectrometers, chromatic dispersion, pulse compression, spectral beam combining
and other articles in the category general optics

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