# Zone Plate Equations

## Zone Plate Equations

Variables

• f = focal length
• λ = wavelength
• ρr = phase variation
• rm = radius of mth zone
• m = zone number
• 1 ≤ N ≤ ∞

Definitions

• GZP = Geometric Zone Plate, paraxial, f# > 10
• IZP = Interferometric Zone Plate, f# < 10
• GHZP = Generalized Holographic Zone Plate

Equations

• For GZP: ${\displaystyle r_{m}={\sqrt {2m\lambda f}},\,\phi _{r}=\pi r^{2}/(\lambda f)}$
• For IZP: ${\displaystyle r_{m}={\sqrt {2m\lambda f+{(m\lambda )}^{2}}}}$
• For GHZP: ${\displaystyle r_{m}={\sqrt {2m\lambda f+{(m\lambda /N)}^{2}}}}$

## Properties of Zone Plates

• ${\displaystyle {\text{Minimum zone width }}\simeq 2\lambda f\sharp \simeq {\text{ minimum spot size}}}$
• ${\displaystyle {\text{Depth of focus }}\simeq 2\lambda {f\sharp }^{2}}$
• ${\displaystyle f={\frac {f_{m}^{2}}{2m\lambda }},\,f\sharp ={\frac {1}{2NA}}={\frac {f}{Dia}}}$
• ${\displaystyle {\text{Chromatic aberration }}\Delta \lambda \simeq \lambda /m}$ (m = number of zones illuminated)
• Hybrid condition (f zone)/(f lens) = (n-1)/λ (dispers const.))
• Hybrid Achromats typically use GZP pattern and less than 300 zones