Physics:Smith–Helmholtz invariant
In optics the Smith–Helmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system. Given an object at height [math]\displaystyle{ \bar{y} }[/math] and an axial ray passing through the same axial position as the object with angle [math]\displaystyle{ u }[/math], the invariant is defined by[1][2][3]
- [math]\displaystyle{ H = n\bar{y}u }[/math],
where [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ i }[/math]th conjugate image point with height [math]\displaystyle{ \bar{y}_i }[/math] and refracted axial ray with angle [math]\displaystyle{ u_i }[/math] in medium with index of refraction [math]\displaystyle{ n_i }[/math] we have [math]\displaystyle{ H = n_i \bar{y}_i u_i }[/math]. 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 [math]\displaystyle{ n u / n' u' }[/math] is constant, where [math]\displaystyle{ u }[/math] and [math]\displaystyle{ n }[/math] are the axial ray angle and refractive index in object space and [math]\displaystyle{ u' }[/math] and [math]\displaystyle{ n' }[/math] are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification, [math]\displaystyle{ y/y' }[/math] 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 [math]\displaystyle{ y'/y }[/math] and [math]\displaystyle{ u'/u }[/math] respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by[5]
- [math]\displaystyle{ \frac{y'}{y} \frac{u'}{u} = \frac{n}{n'} }[/math]
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.
See also
References
- ↑ Born, Max; Wolf, Emil. Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (6th ed.). Pergamon Press. pp. 164-166. ISBN 978-0-08-026482-0.
- ↑ "Technical Note: Lens Fundamentals". https://www.newport.com/n/optics-fundamentals. Retrieved 16 April 2020.
- ↑ Kingslake, Rudolf (2010). Lens design fundamentals (2nd ed.). Amsterdam: Elsevier/Academic Press. pp. 63-64. ISBN 9780819479396.
- ↑ Jenkins, Francis A.; White, Harvey E.. Fundamentals of optics (4th ed.). McGraw-Hill. pp. 173-176. ISBN 0072561912.
- ↑ Born, Max; Wolf, Emil. Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (6th ed.). Pergamon Press. pp. 164-166. ISBN 978-0-08-026482-0.
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