diffraction

Diffraction basics

The diffraction phenomenom is described by the Huygens-Fresnel principle assuming that every point of a surface illuminated by a light wave can be considered as a secondary source emitting a spherical wave ( in all directions ). All these spherical waves (wavelets) interfere to generate the diffracted wave. A consequence of this principle is that a wave going through an aperture (called diffracted wave) has different phase and amplitude distribution than the incident wave. The phase and amplitude distribution of the diffracted wave depend on the phase and amplitude distribution of the incident wave, on the wavelength, on the aperture shape and dimensions. As an example, a plane wave obturated by an aperture becomes diverging. The smaller the aperture, the larger the divergence. Thus, the irradiance of a plane wave diffracted by a circular aperture and focused on a screen located in the focal plane of a lens is not punctual as stated by the geometrical optics theory. Therefore, the diffraction is an important effect that limits the resolution of any optical system.

$diffraction pattern by a circular aperture$ $diffraction pattern by a circular aperture$ The two pictures nearby show the intensity distribution at infinity (or in the focal plane of a lens) of a plane monochromatic wave diffracted respectively by a circular aperture (left picture) and a rectangular aperture (right picture). Without diffraction, the irradiance pattern should be punctual. As real waves have finite dimensions, plane waves do not exist and all waves have a non null divergence.

$diffraction basics$ Based on the Huygens-Fresnel principle, the complex amplitude of a wave at a point P of the space is the superposition of the wavelets coming from all points of any surface S crossed by the wave. This complex amplitude is given by the following formula :

$diffraction formula$ .

M is a point on the surface, ϕ_a the complex amplitude of the wave at M, k the wave vector, dS a surface element around M, d the distance from M to P and Q_H, a factor (called inclination coefficient) inversely proportionnal to a distance. This formula is in general not easy to use. There are cases where it can be simplified using some approximations.

Fraunhofer diffraction

$diffraction basics$ The Fraunhofer diffraction is the particular case for which the wave is monochromatic and the aperture is plane. The surface S mentionned above is then the aperture plane and the point P is located in a plane parallel to the aperture. d₀ is the distance between the aperture center O and P. The Fraunhofer approximation applies when d₀ is much higher than the other parameters. Then the Q_H factor can be considered as constant and the formula can be approximated as follows:

$diffraction formula$ .

The complex amplitude can be expressed with the wavelength λ and the angles θ_x and θ_y illustrated in the graphic above :

$diffraction formula$ .

The intensity is then $diffraction formula$ .

The complex amplitude can also be expressed with the parameters u_x and u_y called the spatial frequencies respectively along the X and Y axis (see MTF tutorial) :

$diffraction formula$ with u_x and u_y defined as follows : $diffraction formula$

For a plane wave propagating perpendicularly to the aperture, the complex amplitude on the aperture is given by its transmittance t(x,y). For example, when the aperture is a simple diaphragm, t(x,y) = 0 around the diaphragm and t(x,y) = 1 inside the diaphragm : in this case, the phase is not changed by the aperture. The simplest cases include a circular or a rectangular diaphragm.

For a rectangular aperture with dimensions a and b respectively along the x and y axis, the normalized intensity is then given by the formula :

$diffraction formula$

For a circular aperture with a diameter D, the intensity has a symetry of revolution around the propagation axis and depends on the angle θ made by the vector OP with this axis :

$diffraction formula$ (J1 is the Bessel function of the first kind).

$diffraction basics$ In both cases, the intensity distribution is made of a central pic (for θ = 0) with attenuated lobes as shown on the figure. One can define the diffracted wave divergence by the angular distance between the two first nodes around the pic as most of the energy is contained inside.

For a circular aperture, the divergence is the same whatever the plane (XZ or YZ, Z being the propagation axis). For a rectangular aperture, the divergence in the XZ plane depends on the dimension a and the divergence in the YZ plane depends on the dimension b. In the limit case where the rectangular aperture is a slit, the wave is only diffracted in the plane perpendicular to the slit axis.

$diffraction basics$ All the optical systems are obturated by an aperture. It can be for example a physical diaphragm like an iris diaphragm used in a camera or a surface of a lens. In general, this aperture is circular. The diffraction by this aperture can be assumed to a Fraunohofer diffraction (at least in the paraxial conditions) and the image of an on-axis point by the optical system (assuming that the optical system is perfect) is not a point but a spot called Airy spot which intensity pattern in the image plane is similar to the intensity pattern observed at infinity (see the figure on the left representing the intensity profile in the image plane). It is given by the formula : $diffraction formula$

r is the distance to the axis in the image plane and z the distance from the secondary principal plane to the image plane. This formula applies only in paraxial conditions and thus for r much smaller than z. As for the diffraction observed at infinity, the first minima of intensity around the pic circumscribe a disk called Airy disk inside which most of the energy is concentrated. The Airy disk diameter is approximately :

$diffraction formula$ where NA_i is the numerical aperture in the image space. The Airy disk diameter can also be calculated with both the focal length of the lens and the aperture diameter in the image space instead of the numerical aperture.

$diffraction basics$ Practically, whatever the optical system quality, it is not possible to obtain an image from a point smaller than the Airy disk. Thus the diffraction is the ultimate resolution limitation of an optical system. The power of resolution is commonly given by D_Airy / 2 (Rayleigh criterion). It is considered as the minimum distance between two Airy spots so that they can be distinguished by the human eye. The left figures illustrate the normalized intensity (green curve) obtained by summing the normalized intensity of two Airy spots at a distance from each other of D_Airy / 2 (blue and red curves).

Diffraction by a grating

A grating is substrate whith a periodic succesion of diffractive elements. Thus, an incident light wave is scattered by each diffractive element and the emerging wave is the result of the interference between each scattered wave. A grating can be transmissive or reflective depending if the diffractive elements spread the light in the same sense or in the opposite sense than the incident wave. The most common gratings are plane gratings with a flat surface where the diffractive elements are arranged in parallel lines with a constant distance between them. The simplest diffractive elements are slits with a uniform transmittance. In the case of a plane monochromatic incident wave, the aforesaid intererferences are constructive in discret directions only. Therefore, the emerging wave is the superposition of plane waves with different directions corresponding to the constructive interference. These directions depend among other on the wavelength. Accordingly, a polychromatic plane wave is diffracted in as many directions as wavelengths. In other words, the grating allows to spread spatially the incident wave spectrum. Thats the reason why gratings are very common components in spectrometers. They are also used in many other devices as spectral filters in laser cavities, pulse compressors/stretchers in short pulse lasers amplifiers, etc.

Constructive interference occure when the phase difference Δφ induced between two consecutive elements is a multiple of 2π. The formulas below give the emerging direction id depending on the incident direction i , the wavelength λ (the wave incident wave is supposed plane and monochromatic), the pitch d (space between 2 diffractive elements) and the order k (The order k is the ratio Δφ/2π):

$diffraction by a transmissive grating$ transmissive grating:

$diffraction by a grating$

$diffraction by a reflective grating$ reflective grating:

$diffraction by a grating$

k can be either positive or negative. The order 0 corresponds to the direction of the specular reflexion (reflective grating) or the direction transmitted by the substrate (transmissive grating). Note that depending on the grating efficiency, more or less intensity remains in the 0 order direction.

$diffraction by a reflective grating$ A more complete analysis of the diffracted wave can be made easily in the case where the diffractive elements are slits with a uniform transmittance. Each slit diffracts the incident wave according to the following formula : $diffraction by a grating$ where Φ_j is the complex amplitude of the slit number j and u is defined as follows : $diffraction by a grating$ , i_e being the observation angle after diffraction. The complex amplitude Φ of the diffracted wave is obtained by summing the complex amplitudes of the waves diffracted by each single slit : $diffraction by a grating$ .

$diffraction by a reflective grating$ The normalized intensity of the diffracted wave is then :

$diffraction by a grating$ where N = 2.n+1 is the total number of slits and dx is the slits width. The normalized intensity is illustrated on the graphic nearby by the blue curve which is enveloped by the red dashed curve corresponding to the relative intensity obtained after diffraction by a single slit. Each pic corresponds to an order of diffraction, the order 0 (at the center of the curve) being in the present case the most efficient. Note that the envelope shape and its position depend on the diffractive elements that can be designed for maximizing the intensity diffracted in a given order.

The curve shows that the wider the grating, the thinner the diffracted order. Indeed, the Full Width Half Maximum (FWHM) is $diffraction by a grating$ where D is the wave width (or the grating width if the wave is larger than the grating). Also, the thinner the slits, the flatter the envelope which means that the intensity is more uniformly distributed among the different orders.

In the general case, one can define the angular dispersion for a certain order k is $diffraction by a grating$ . One important parameter is the resolvance Res which is defined as the ratio between the wavelength λ and the smallest resolvable wavelength variation dλ.
dλ is defined according the Rayleigh criterion and corresponds to the FWHM (Full Width Half Maximum) of the diffracted pic. Therefore :

$diffraction formula$ . The higher the order, the larger the resolvance. However, low diffraction orders are in general brighter than high diffraction orders.

Another important parameter is the free spectral range of the grating $diffraction formula$ , λ being the smallest wavelength of the spectrum. It is the maximum spectral width of a plane wave for avoiding the spatial overlay of the considered diffraction order with the adjacent diffraction orders.

Littrow angle

$diffraction by a reflective grating$ There is a special angle named Littrow angle for which the angle of incidence is the same than the diffracted angle in the first order. Its value is given in the following formula :
$diffraction formula$ .
Gratings are usually used in spectrally tunable external cavity lasers. In these devices, a laser diode with an anti reflexion coating on its front end facet is used as an amplifier. The beam coming from the amplifier is collimated and hits a grating oriented with Littrow angle for a given wavelength λ_l. The laser cavity is made of the laser diode back end facet and the grating. Only the diffracted wave in the first order at λ_l is reflected back in the amplifier and can oscillate. by rotating the grating, the incidence angle corresponds to the Littrow angle of another wavelength which can then oscillate. It is then possible to tune the wavelength emitted by rotating the grating.

References

"Optique Fondements et applications" - 2004 - author : José-Philippe Perez.

"Cours doptique physique" - Institut doptique théorique et appliquée - 1985 - author : Christian Imbert.

"Cours doptique ondulatoire" - Université Denis Diderot Paris 7 - 2006 - author : G.Rebmann.

"Études graphiques des propriétés optiques des lames minces" - Journal de physique - 1950 - author : D. Malé.