Spatial filter
Encyclopedia
A spatial filter is an optical device which uses the principles of Fourier optics
to alter the structure of a beam of coherent
light
or other electromagnetic radiation
. Spatial filtering is commonly used to "clean up" the output of laser
s, removing aberrations in the beam due to imperfect, dirty, or damaged optics, or due to variations in the laser gain medium itself. This can be used to produce a laser beam containing only a single transverse mode
of the laser's optical resonator.
In spatial filtering, a lens
is used to focus
the beam. Because of diffraction
, a beam that is not a perfect plane wave
will not focus to a single spot, but rather will produce a pattern of light and dark regions in the focal plane. For example, an imperfect beam might form a bright spot surrounded by a series of concentric rings, as shown in the figure to the right. It can be shown that this two-dimensional pattern is the two-dimensional Fourier transform
of the initial beam's transverse intensity
distribution. In this context, the focal plane is often called the transform plane. Light in the very center of the transform pattern corresponds to a perfect, wide plane wave. Other light corresponds to "structure" in the beam, with light further from the central spot corresponding to structure with higher spatial frequency
. A pattern with very fine details will produce light very far from the transform plane's central spot. In the example above, the large central spot and rings of light surrounding it are due to the structure resulting when the beam passed through a circular aperture
. The spot is enlarged because the beam is limited by the aperture to a finite size, and the rings relate to the sharp edges of the beam created by the edges of the aperture. This pattern is called an Airy pattern, after its discoverer George Airy
.
By altering the distribution of light in the transform plane and using another lens to reform the collimated beam, the structure of the beam can be altered. The most common way of doing this is to place an aperture in the beam that allows the desired light to pass, while blocking light that corresponds to undesired structure in the beam. In particular, a small circular aperture or "pinhole" that passes only the central bright spot can remove nearly all fine structure from the beam, producing a smooth transverse intensity profile, which may be almost a perfect gaussian beam
. With good optics and a very small pinhole, one could even approximate a plane wave.
In practice, the diameter of the aperture is chosen based on the focal length
of the lens, the diameter and quality of the input beam, and its wavelength
(longer wavelengths require larger apertures). If the hole is too small, the beam quality is greatly improved but the power
is greatly reduced. If the hole is too large, the beam quality may not be improved as much as desired.
The size of aperture that can be used also depends on the size and quality of the optics. To use a very small pinhole, one must use a focusing lens with a low f-number
, and ideally the lens should not add significant aberrations
to the beam. The design of such a lens becomes increasingly more difficult as the f-number decreases.
In practice, the most commonly used configuration is to use a microscope objective lens for focusing the beam, and an aperture made by punching a small, precise, hole in a piece of thick metal foil. Such assemblies are available commercially.
wavefront can be produced. As with plane waves, a smaller aperture corresponds to a better approximation of a spherical wavefront.
Fourier optics
Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...
to alter the structure of a beam of coherent
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....
light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
or other electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
. Spatial filtering is commonly used to "clean up" the output of laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
s, removing aberrations in the beam due to imperfect, dirty, or damaged optics, or due to variations in the laser gain medium itself. This can be used to produce a laser beam containing only a single transverse mode
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
of the laser's optical resonator.
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...
is used to focus
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
the beam. Because of diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
, a beam that is not a perfect plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
will not focus to a single spot, but rather will produce a pattern of light and dark regions in the focal plane. For example, an imperfect beam might form a bright spot surrounded by a series of concentric rings, as shown in the figure to the right. It can be shown that this two-dimensional pattern is the two-dimensional Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the initial beam's transverse intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
distribution. In this context, the focal plane is often called the transform plane. Light in the very center of the transform pattern corresponds to a perfect, wide plane wave. Other light corresponds to "structure" in the beam, with light further from the central spot corresponding to structure with higher spatial frequency
Spatial frequency
In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is...
. A pattern with very fine details will produce light very far from the transform plane's central spot. In the example above, the large central spot and rings of light surrounding it are due to the structure resulting when the beam passed through a circular aperture
Aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane. The aperture determines how collimated the admitted rays are,...
. The spot is enlarged because the beam is limited by the aperture to a finite size, and the rings relate to the sharp edges of the beam created by the edges of the aperture. This pattern is called an Airy pattern, after its discoverer George Airy
George Biddell Airy
Sir George Biddell Airy PRS KCB was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881...
.
By altering the distribution of light in the transform plane and using another lens to reform the collimated beam, the structure of the beam can be altered. The most common way of doing this is to place an aperture in the beam that allows the desired light to pass, while blocking light that corresponds to undesired structure in the beam. In particular, a small circular aperture or "pinhole" that passes only the central bright spot can remove nearly all fine structure from the beam, producing a smooth transverse intensity profile, which may be almost a perfect gaussian beam
Gaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity distributions are well approximated by Gaussian functions. Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental...
. With good optics and a very small pinhole, one could even approximate a plane wave.
In practice, the diameter of the aperture is chosen based on the focal length
Focal length
The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...
of the lens, the diameter and quality of the input beam, and its wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
(longer wavelengths require larger apertures). If the hole is too small, the beam quality is greatly improved but the power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
is greatly reduced. If the hole is too large, the beam quality may not be improved as much as desired.
The size of aperture that can be used also depends on the size and quality of the optics. To use a very small pinhole, one must use a focusing lens with a low f-number
F-number
In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...
, and ideally the lens should not add significant aberrations
Aberration in optical systems
Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...
to the beam. The design of such a lens becomes increasingly more difficult as the f-number decreases.
In practice, the most commonly used configuration is to use a microscope objective lens for focusing the beam, and an aperture made by punching a small, precise, hole in a piece of thick metal foil. Such assemblies are available commercially.
Spherical waves
By omitting the second lens that reforms the collimated beam, light that approximates a sphericalSphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
wavefront can be produced. As with plane waves, a smaller aperture corresponds to a better approximation of a spherical wavefront.