optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

the Smith–Helmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system. Given an object at height

\bar

and an axial ray passing through the same axial position as the object with angle

u

, the invariant is defined by :

H = n\baru

, where

n

is the

refractive index In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...

. For a given optical system and specific choice of object height and axial ray, this quantity is invariant under

refraction In physics, refraction is the redirection of a wave as it passes from one transmission medium, medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commo ...

. Therefore, at the

i

th conjugate image point with height

\bar_i

and refracted axial ray with angle

u_i

in medium with index of refraction

n_i

we have

H = n_i \bar_i u_i

. Typically the two points of most interest are the object point and the final image point. The Smith–Helmholtz invariant has a close connection with the Abbe sine condition. The paraxial version of the sine condition is satisfied if the ratio

n u / n' u'

is constant, where

u

and

n

are the axial ray angle and refractive index in object space and

u'

and

n'

are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification,

y/y'

is constant if and only if the sine condition is satisfied. The Smith–Helmholtz invariant also relates the lateral and angular magnification of the optical system, which are

y'/y

and

u'/u

respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by :

\frac \frac = \frac

The Smith–Helmholtz invariant is closely related to the Lagrange invariant and the optical invariant. The Smith–Helmholtz is the optical invariant restricted to conjugate image planes.

References

{{DEFAULTSORT:Smith-Helmholtz invariant Geometrical optics

See also

References