Herschel's condition

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.^[1]

The Herschel's condition in mathematical form is $${\displaystyle {\frac {1-\cos \alpha _{\mathrm {o} }}{1-\cos \alpha _{\mathrm {i} }}}={\frac {1-\cos \beta _{\mathrm {o} }}{1-\cos \beta _{\mathrm {i} }}}={\frac {n_{i}^{2}}{n_{o}^{2}}}|M_{T}|^{2}}$$ where ${\displaystyle \alpha _{o},\beta _{o}}$ are the object side ray angle, ${\displaystyle \alpha _{i},\beta _{i}}$ are the image side ray angle. ${\displaystyle n_{o},n_{i}}$ are the object and image side refractive index, and ${\displaystyle M_{T}}$ is the transverse magnification. This condition can be derived by the Fermat's principle.^[2]

This condition can also be expressed as^{[3]: §1.9 [4]: §29.8} $${\displaystyle M_{L}={\frac {n_{o}\sin ^{2}(\alpha _{\mathrm {o} }/2)}{n_{i}\sin ^{2}(\alpha _{\mathrm {i} }/2)}}={\frac {n_{o}(1-\cos \alpha _{\mathrm {o} })}{n_{i}(1-\cos \alpha _{\mathrm {i} })}}{\text{ and }}M_{T}={\frac {n_{o}\sin(\alpha _{\mathrm {o} }/2)}{n_{i}\sin(\alpha _{\mathrm {i} }/2)}}}$$ where ${\displaystyle M_{L}={\frac {n_{i}}{n_{o}}}M_{T}^{2}}$ 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 ${\displaystyle |M_{T}|=n_{o}/n_{i}}$.^[3]: §1.9

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