In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, vergence is the angle formed by
rays
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (gra ...
of light that are not perfectly
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
to one another. Rays that move closer to the
optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light propaga ...
as they propagate are said to be ''converging'', while rays that move away from the axis are
''diverging''. These imaginary rays are always perpendicular to the
wavefront
In physics, the wavefront of a time-varying '' wave field'' is the set ( locus) of all points having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal fr ...
of the light, thus the vergence of the light is directly related to the
radii of curvature of the wavefronts. A convex
lens
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
or concave
mirror
A mirror or looking glass is an object that reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the direction of the im ...
will cause parallel rays to focus, converging toward a point. Beyond that
focal point, the rays diverge. Conversely, a concave lens or convex mirror will cause parallel rays to diverge.
Light does not actually consist of imaginary rays and light sources are not single-point sources, thus vergence is typically limited to simple ray modeling of optical systems. In a real system, the vergence is a product of the diameter of a light source, its distance from the optics, and the curvature of the optical surfaces. An increase in curvature causes an increase in vergence and a decrease in
focal length, and the image or spot size (waist diameter) will be smaller. Likewise, a decrease in curvature decreases vergence, resulting in a longer focal length and an increase in image or spot diameter. This reciprocal relationship between vergence, focal length, and waist diameter are constant throughout an optical system, and is referred to as the
optical invariant
In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by
:H = n\overliney - nu\overline,
where and are the marginal ray height and angle respectively, and and are the chief ray heigh ...
. A beam that is expanded to a larger diameter will have a lower degree of divergence, but if condensed to a smaller diameter the divergence will be greater.
The simple ray model fails for some situations, such as for
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
light, where
Gaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. Thi ...
analysis must be used instead.
Definition
![PSM V28 D191 Light wave divergence traveling through different focal points](https://upload.wikimedia.org/wikipedia/commons/6/6e/PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg)
In
geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of '' rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
, vergence describes the
curvature of optical wavefronts.
[ ] Vergence is defined as
:
where ''n'' is the medium's refractive index and ''r'' is the distance from the point source to the wavefront. Vergence is measured in units of
dioptres (D) which are equivalent to m
−1.
This describes the vergence in terms of
optical power
In optics, optical power (also referred to as dioptric power, refractive power, focusing power, or convergence power) is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the ...
. For optics like convex lenses, the converging point of the light exiting the lens is on the input side of the
focal plane, and is positive in optical power. For concave lenses, the
focal point is on the back side of the lens, or the output side of the focal plane, and is negative in power. A lens with no optical power is called an
optical window
The optical window is a range of wavelengths that are not blocked by the earth's atmosphere. The window runs from around 300 nanometers (ultraviolet-B) up into the range the human eye can detect, roughly 400–700 nm and continues up to approxim ...
, having flat, parallel faces. The optical power directly relates to how large positive images will be magnified, and how small negative images will be diminished.
All light sources produce some degree of divergence, as the waves exiting these sources always have some degree of curvature. At the proper distance, these waves can be straightened by use of a lens or mirror, creating
collimated
A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction p ...
beams with minimal divergence, but some degree of divergence will remain, depending on the diameter of the beam versus the focal length.
[''Ophthalmic Medical Personnel: A Guide to Laws, Formulae, Calculations, and Clinical Applications'' by Aaron V. Shukla – Slack Inc. 2009 Page 73–76] When the distance between the point source and wavefront becomes very large, the vergence goes to zero meaning that the wavefronts are planar and no longer have any detectable curvature. Light from distant stars have such a large radius that any curvature of the wavefronts are undetectable, and have no vergence.
[
The light can also be pictured as consisting of a bundle of lines radiating in the direction of propagation, which are always perpendicular to the wavefront, called "rays". These imaginary lines of infinitely small thicknesses are separated by only the angle between them. In ray tracing, the vergence can then be pictured as the angle between any two rays. For imaging or beams, the vergence is often described as the angle between the outermost rays in the bundle (]marginal ray
In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation ...
s), at the edge (verge) of a cone of light, and the optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light propaga ...
. This slope is typically measured in radians
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
. Thus, in this case the convergence of the rays transmitted by a lens is equal to the radius of the light source divided by its distance from the optics. This limits the size of an image or the minimum spot diameter that can be produced by any focusing optics, which is determined by the reciprocal of that equation; the divergence of the light source multiplied by the distance. This relationship between vergence, focal length, and the minimum spot diameter (also called the "waist diameter") remains constant through all space, and is commonly referred to as the optical invariant
In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by
:H = n\overliney - nu\overline,
where and are the marginal ray height and angle respectively, and and are the chief ray heigh ...
.
This angular relationship becomes especially important with laser operations such as laser cutting or laser welding
Laser beam welding (LBW) is a welding technique used to join pieces of metal or thermoplastics through the use of a laser. The beam provides a concentrated heat source, allowing for narrow, deep welds and high welding rates. The process is frequen ...
, as there is always a trade off between spot diameter, which affects the intensity of the energy, and the distance to the object. When low divergence in the beam is desired, then a larger diameter beam is necessary, but if a smaller beam is needed one must settle for greater divergence, and no change in the position of the lens will alter this. The only way to achieve a smaller spot is to use a lens with a shorter focal-length, or expand the beam to a larger diameter.
However, this measure of the curvature of wavefronts is only fully valid in geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of '' rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
, not in Gaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. Thi ...
optics or in wave optics
In physics, physical optics, or wave optics, is the branch of optics that studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. This usage tends not to include effe ...
, where the wavefront at the focus is wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
-dependent and the curvature is not proportional to the distance from the focus. In this case, the diffraction of the light starts to play a very active role, often limiting the spot size to even larger diameters, especially in the far field
The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ante ...
. For non-circular light sources, the divergence may differ depending on the cross-sectional position of the rays from the optical axis. Diode laser
The laser diode chip removed and placed on the eye of a needle for scale
A laser diode (LD, also injection laser diode or ILD, or diode laser) is a semiconductor device similar to a light-emitting diode in which a diode pumped directly with e ...
s, for example, have greater divergence across the parallel direction (fast axis) than the perpendicular (slow axis), producing beams with rectangular profiles. This type of difference in divergence can be reduced by beam-shaping methods, such as using a rod lens that only affects divergence along a single cross-sectional direction.[''Laser Beam Shaping Applications'' by Fred M. Dickey, Todd E. Lizotte – CRC Press 2017 Page 76–77]
Convergence, divergence, and sign convention
Wavefronts propagating toward a single point yield positive vergence. This is also referred to as ''convergence'' since the wavefronts are all converging to the same point of focus. Contrarily, wavefronts propagating away from a single source point give way to negative vergence. Negative vergence is also called ''divergence''.
See also
*Optical power
In optics, optical power (also referred to as dioptric power, refractive power, focusing power, or convergence power) is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the ...
* Optical resolution
*Cardinal point (optics)
In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' s ...
References
{{DEFAULTSORT:Vergence (Optics)
Geometrical optics