X-Ray Optics Calculator

• Refractive Optics
Main Formulae
• Refraction Index decrement
• Compound Refractive Lens (cylindrical)
• Compound Refractive Lens (parabolical)
• Planar Parabolic Refractive Lens
• Planar Kinoform Refractive Lens

Main Formulae

Optical properties versus material properties and wavelength

• Linear attenuation coefficient μ
        μ = 4 π β / λ
   
• Length of 2π phase shift L_2π
        L_2π = λ/ δ
   
• Transmission through length of 2π phase shift L_2π
        T_2π = exp(- λ μ / δ)  = exp(- 4 π β / δ)
   
• Length of π phase shift L_π
        L_π = λ/ 2δ
   
• Transmission through length of π phase shift L_π
        T_π = exp(- λ μ / 2δ)  = exp(- 2 π β / δ)
   
• Length of π/2 phase shift L_π/2
        L_π/2 = λ/ 4δ
   
• Transmission through length of π/2 phase shift L_π/2
        T_π/2 = exp(- λ μ / 4δ)  = exp(-  π β / δ)
   
• Effective aperture A_eff
        A_eff = (f λ δ / β)^½  = (4 π f δ / μ)^½
   
• Diffraction-limited resolution for plane wave σ
        σ = 0.664 (f λ β / δ)^½  = 0.664 [f λ² μ / (4 π δ)]^½
   
• Minimal radius of curvature at parabola apex R_C
        R_C = 2 f δ
   
• Tolerance on radius of curvature at parabola apex ΔR_C/R_C
       ΔR_C/R_C =  (6.4/π) (β / δ) =  (1.6/π²) λ(μ / δ)
   
• Transmission T_A through full aperture
        T_A = [√(fλδ/(2β)) / A]  erf[A /  (2√(fλδ/(2πβ))]
   
• Transmission T through effective aperture A_eff
        T = (1/√2) erf(√(π/2))  ≈ 0.65314 = 65.314%
   
• Maximum possible gain G
        G = T A_eff / σ  ≈ δ/β  = 4 π δ / (λ μ)