Herschel's condition

Entrance and exit rays through an imaging system (grey box).

In optics, the Herschel's condition is a condition for an optical system to produce sharp images for objects over an extended axial range, i.e. for objects displaced along the optical axis. It was formulated by John Herschel.

Mathematical formulation

The Herschel's condition in mathematical form is $$\frac{1-\cos {\alpha }_{\mathrm{o}}}{1-\cos {\alpha }_{\mathrm{i}}}=\frac{1-\cos {\beta }_{\mathrm{o}}}{1-\cos {\beta }_{\mathrm{i}}}=\frac{n_i^2}{n_o^2}|M_T{|}^2$$ where ${\alpha }_o,{\beta }_o$ are the object side ray angles, ${\alpha }_i,{\beta }_i$ are the image side ray angle. $n_o,n_i$ are the object and image side refractive index, and $M_T$ is the transverse magnification. This condition can be derived by the Fermat's principle.

This condition can also be expressed asTemplate:RpTemplate:Rp $$M_L=\frac{n_o{\sin }^2({\alpha }_{\mathrm{o}}/2)}{n_i{\sin }^2({\alpha }_{\mathrm{i}}/2)}=\frac{n_o(1-\cos {\alpha }_{\mathrm{o}})}{n_i(1-\cos {\alpha }_{\mathrm{i}})}\text{ and }{M}_T=\frac{n_o\sin ({\alpha }_{\mathrm{o}}/2)}{n_i\sin ({\alpha }_{\mathrm{i}}/2)}$$ where $M_L=\frac{n_i}{n_o}{M}_T^2$ is the longitudinal magnification.

This condition is in general conflict with the Abbe sine condition, which is the condition for aberration free imaging for objects displaced off-axis. They can be simultaneously satisfied only when the system has magnification equal to the ratio of refractive index $|M_T|=n_o/n_i$.Template:Rp

See also

• Lagrange invariant
• Smith-Helmholtz invariant
• Abbe sine condition