Smith–Helmholtz invariant explained

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

\bar{y}

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

, the invariant is defined by^[1] ^[2] ^[3]

H=n\bar{y}u

,where

is the refractive index. For a given optical system and specific choice of object height and axial ray, this quantity is invariant under refraction. Therefore, at the

th conjugate image point with height

\bar{y}_i

and refracted axial ray with angle

u_i

in medium with index of refraction

n_i

we have

H=n_i\bar{y}_iu_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

nu/n'u'

is constant, where

and

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

and

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.^[4]

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^[5]

	y'
	y

	u'
	u

	n
	n'

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.

Notes and References

Book: Born . Max . Wolf . Emil . . Pergamon Press . 978-0-08-026482-0 . 164-166 . 6th.
Web site: Newport . Technical Note: Lens Fundamentals . 16 April 2020.
Book: Kingslake . Rudolf . Lens design fundamentals . 2010 . Elsevier/Academic Press . Amsterdam . 9780819479396 . 63-64 . 2nd.
Book: Jenkins . Francis A. . White . Harvey E. . Fundamentals of optics . McGraw-Hill . 0072561912 . 173-176 . 4th.
Book: Born . Max . Wolf . Emil . . Pergamon Press . 978-0-08-026482-0 . 164-166 . 6th.

Smith–Helmholtz invariant explained

See also

Notes and References