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Small angle approximation in geometric optics
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system
Paraxial_approximation
Eigenvalue problem for the Laplace operator
diffraction theory, e.g. in deriving Fresnel diffraction. In the paraxial approximation of the Helmholtz equation, the complex amplitude A is expressed
Helmholtz_equation
Simplification of the basic trigonometric functions
optics, the small-angle approximations form the basis of the paraxial approximation. The sine and tangent small-angle approximations are used in relation
Small-angle_approximation
Study of classical optics using Fourier transforms
\over c}} is the wave number. Next, use the paraxial approximation, that is a small-angle approximation such that k x 2 + k y 2 ≪ k z 2 {\displaystyle
Fourier_optics
Idealized model of light
reasonably well by using the paraxial approximation. When discussing ray tracing this definition is often reversed: a "paraxial ray" is then a ray that is
Ray_(optics)
Monochrome light beam whose amplitude envelope is a Gaussian function
§ Complex conjugate ambiguity. Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above
Gaussian_beam
Approximation of a function by its tangent line at a point
the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric
Linear_approximation
Lens with a thickness that is negligible
approximation ignores optical effects due to the thickness of lenses and simplifies ray tracing calculations. It is often combined with the paraxial approximation
Thin_lens
Optical device which transmits and refracts light
&={\frac {h}{R}}\end{aligned}}} , and using small angle approximation (paraxial approximation) and eliminating i, r, and θ, n 2 v + n 1 u = n 2 − n 1
Lens
Ray tracing technique
optics. This technique, as described below, is derived using the paraxial approximation, which requires that all ray directions (directions normal to the
Ray_transfer_matrix_analysis
Technique in geometric optics
the paraxial approximation, in which only rays that make small angles with the optical axis of the system are considered. In this approximation, the
Gaussian_optics
Characteristic of an optical system
{\displaystyle a=2\theta } , is approximately twice this value (within the paraxial approximation). The NA is generally measured with respect to a particular object
Numerical_aperture
Six points which determine imaging properties of an optical system
has on rays that pass through that point, in the paraxial approximation. The paraxial approximation assumes that rays travel at shallow angles with respect
Cardinal_point_(optics)
German polymath and scholar (1777–1855)
formation of images under a paraxial approximation (Gaussian optics). He characterized optical systems under a paraxial approximation only by its cardinal points
Carl_Friedrich_Gauss
Optical phenomenon
\propto e^{im\phi }e^{-r^{2}},\!} is a solution to the paraxial wave equation (see paraxial approximation, and the Fourier optics article for the actual equation)
Optical_vortex
Mirror with a curved reflecting surface
mathematical treatment is done under the paraxial approximation, meaning that under the first approximation a spherical mirror is a parabolic reflector
Curved_mirror
Branch of physics that studies light
Geometric optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behaviour then becomes linear
Optics
Electron trajectories in electromagnetic fields
design of electron microscopes and particle accelerators. In the paraxial approximation, trajectory calculations can be carried out using ray transfer matrix
Electron_optics
Concept in laser optics
{div} }}}} . These equations are valid within the limits of the paraxial approximation. For beams with much larger divergence the Gaussian beam model is
Rayleigh_length
Physics related to the study, design, building and operation of particle accelerators
cases using the Paraxial approximation. Even in the cases of strongly nonlinear magnetic fields, and without the paraxial approximation, a Lie transform
Accelerator_physics
Parameter describing conic sections
−1), and hyperbolic (K < −1) lens and mirror surfaces. When the paraxial approximation is valid, the optical surface can be treated as a spherical surface
Conic_constant
Type of angular momentum in light
modes Laguerre-Gaussian modes Spin angular momentum of light Paraxial approximation Polarization (waves) Siae Microelettronica patent Willner, Alan
Orbital angular momentum of light
Orbital_angular_momentum_of_light
Near-field diffraction
the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation
Fresnel_diffraction
Process of enlarging the apparent size of something
relative to the eye. The angular magnification MA = ε/ε0 can be, in paraxial approximation where tan(ε) ≈ ε, expressed as (hi/Li)/(ho/LN) = (hiLN)/(hoLi) where
Magnification
the transversal profile of the beam. In the quasi-monochromatic paraxial approximation, the gain can be taken into account with the following equation
Gain_(laser)
Nonlinear optical process
realistic condition in practice, especially in biological samples. The paraxial approximation is however supposed still valid: k n = n k 1 {\displaystyle k_{n}=nk_{1}}
Second-harmonic_generation
Measure of laser beam quality
measure of beam quality. The general wave equation, assuming paraxial approximation, yields: B P P = φ ⋅ w 0 = M 2 ⋅ λ π {\displaystyle \mathrm {BPP}
Beam_parameter_product
Distance between the nearest and the furthest objects that are in focus in an image
significant simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography
Depth_of_field
Model of optics describing light as geometric rays
Geometrical optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behavior then becomes linear
Geometrical_optics
{NA} =\sin a/2=\sin \arctan \left({\frac {D}{2f}}\right)} In the paraxial approximation, with a small aperture, D < f {\displaystyle D<f} : N A ≈ a / 2
Angular_aperture
Statement based on repeated empirical observations that describes some natural phenomenon
In geometric optics laws are based on approximations in Euclidean geometry (such as the paraxial approximation). Law of reflection Law of refraction,
Scientific_law
State of matter with properties of both conventional liquids and crystals
liquid crystal layer should be spherical or paraboloidal under paraxial approximation. As for projecting images or sensing objects, it may be expected
Liquid_crystal
Material with a negative refractive index
with surface plasmons. In another direction researchers explored paraxial approximations of NIM slabs. The existence of negative refractive materials can
Negative-index_metamaterial
Wigner distribution function in physics as opposed to in signal processing
p/ħ is replaced with k = |k| sin θ ≈ |k|θ in the small-angle (paraxial) approximation. In this context, the Wigner function is the closest one can get
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Device to reflect radiation back to its source
transparent sphere and (optionally) a spherical mirror. In the paraxial approximation, this effect can be achieved with lowest divergence with a single
Retroreflector
PE were derived from the slowly varying envelope approximation and they are the so-called paraxial one-way models. Since then, a number of improved one-way
Beam_propagation_method
Numerical model of nonlinear optical systems
{\displaystyle \nabla _{\perp }^{2}} describes diffraction in the paraxial approximation. Conditions of self-focusing are assumed. We refer to Eq.(1) as
Lugiato–Lefever_equation
tweezers optical waveguide optical window optics optoelectronics paraxial approximation pattern recognition pentaprism penumbra periscope phase (waves)
Index_of_optics_articles
Modes of vibration in mathematics
pool, as well as to a mode of an idealized optical fiber in the paraxial approximation. The last application is most practical in connection to the double-clad
Dirichlet_eigenvalue
the first systematic analysis of the formation of images under a paraxial approximation (Gaussian optics). Robert Bunsen invents the Bunsen cell. British
1840_in_science
time variations (and the paraxial approximation) is considered as well. By solving the problem with the mentioned approximations a parabolic partial differential
Time-domain_holography
oscillator Parametric resonance Parasitic drag Parastatistics Parawing Paraxial approximation Parfocal lens Parhelic circle Paris' law Pariser–Parr–Pople method
Index_of_physics_articles_(P)
Deviation from perfect paraxial optical behavior
departure of the performance of an optical system from the predictions of paraxial optics. In an imaging system, it occurs when light from one point of an
Optical_aberration
Optical property
evolves noticeably with propagation distance and the Fresnel diffraction (paraxial) integral is commonly used. F ≪ 1 {\displaystyle F\ll 1} is often associated
Fresnel_number
as a chirp multiplication. The parameters are all simplified as paraxial approximations while meeting the freespace propagation. It does not consider aperture
Focus recovery based on the linear canonical transform
Focus_recovery_based_on_the_linear_canonical_transform
Mexican physicist
He has also shown that a GRIN medium, when studied beyond the paraxial approximation, generates an analogy with a quantum Kerr medium. By being able
Héctor_Manuel_Moya_Cessa
Used to understand the Dirac equation
to understand the propagation of the quasi-paraxial beam in terms of a series of approximations (paraxial plus nonparaxial). Similar is the situation
Foldy–Wouthuysen transformation
Foldy–Wouthuysen_transformation
Non-diffractive wave
1088/2040-8978/12/12/124002. S2CID 120332951. Rosen, J.; Yariv, A. (1995). "Snake beam: a paraxial arbitrary focal line". Optics Letters. 20 (20): 2042–4. Bibcode:1995OptL
Bessel_beam
relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Some
Multislice
Plane curve: conic section
Electromagnetism and Optics, lectures. University of Texas at Austin. Paraxial Optics. Retrieved October 5, 2011. Kumpel, P. G. (1975), "Do similar figures
Parabola
optics (in which light is described as a scalar wave, an approximation that works well for paraxial light with uniform polarization), the light-ray field
METATOY
Telescope for observations with visible light
Aberrations. Spherical aberration The difference in focal length between paraxial rays and marginal rays, proportional to the square of the objective diameter
Optical_telescope
Type of optical fiber
through the core, and hence cannot pump it. Ray tracing, simulations of the paraxial propagation and mode analysis give similar results. In general, modes of
Double-clad_fiber
Electric circuit with two pairs of terminals
light waves in transparent layers Ray transfer matrix for calculation of paraxial propagation of a light ray The emitter-leg resistors counteract any current
Two-port_network
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