Equation Summary

Numerical aperture and f /# :

$$NA=n\prime \sin u\prime f/\#=\frac{f}{EPD}$$

Rayleigh criterion:

$$\text{Δ}X=\frac{0.61\text{λ}}{NA}$$

Image height as a function of field angle:

$$h\prime =f\tan \text{θ}$$

Transverse ray error:

$${\epsilon }_y^{\prime }=\frac{1}{n\prime u{\prime }_a}\frac{\partial W}{\partial {\text{ρ}}_y}{\epsilon }_x^{\prime }=\frac{1}{n\prime u{\prime }_a}\frac{\partial W}{\partial {\text{ρ}}_x}$$

Strehl ratio:

$$Strehl\cong \left(1-2{\pi }^2{\omega }^2\right)\omega =RM{S}_{OPD}$$

Wavefront aberration polynomial:

$$\begin{array}{l}{W}_{IJK}\Rightarrow H^I{\text{ρ}}^J{\cos }^K\text{θ}\\ W\left(H,\text{ρ},\text{θ}\right)=W_{020}{\text{ρ}}^2+W_{111}H\text{ρ}\cos \text{θ}+W_{040}{\text{ρ}}^4+W_{131}H{\text{ρ}}^3\cos \text{θ}\\ +W_{222}{H}^2{\text{ρ}}^2{\cos }^2\text{θ}+W_{220}{H}^2{\text{ρ}}^2+W_{311}{H}^3\text{ρ}\cos \text{θ}+O(6)\end{array}$$

Contrast:

$$Contrast=\frac{I_{\text{max}}-I_{\text{min}}}{I_{\text{max}}+I_{\text{min}}}$$

Focal lengths of any two thin lens system:

$$f_a=\frac{df}{f-BFL}{f}_b=\frac{dBFL}{f-BFL-d}$$

Zero-Petzval solution for two thin lenses:

$$f_a=-f_b=f-BFLd=\frac{{\left(f-BFL\right)}^2}{f}$$

Two-mirror solution:

$$c_1=\frac{BFL-f}{2df}{c}_2=\frac{BFL+d-f}{2dBFL}$$

Schwarzchild solution:

$$d=2f{c}_1=\left(\sqrt{5}-1\right)fc2=\left(\sqrt{5}+1\right)f$$

Aplanatic condition:

$$i=-u^{\prime }$$

Bending and shape factors:

$$\text{β}=\frac{c_1+c_2}{c_1-c_2}C=\frac{u_a+u_a^{\prime }}{u_a-u_a^{\prime }}$$

Thin lens bending for minimum spherical:

$$\frac{c_2}{c_1}=\frac{2n^2-n-4}{n\left(2n+1\right)}$$

Thin lens bending for minimum coma:

$$\frac{c_2}{c_1}=\frac{\left(n^2-n-1\right)}{n^2}$$

Achromatic doublet:

$$\text{Φ}={\text{ϕ}}_1+{\text{ϕ}}_2{\text{ϕ}}_1=\text{Φ}\frac{V_1}{V_1-V_2}{\text{ϕ}}_2=-\text{Φ}\frac{V_2}{V_1-V_2}$$

Petzval sum:

$${\displaystyle \sum _j\frac{{\text{ϕ}}_j}{n_j}}$$

Thick lens power:

$$\text{ϕ}={\text{ϕ}}_1+{\text{ϕ}}_2-\frac{t}{n}\left({\text{ϕ}}_1{\text{ϕ}}_2\right)$$

Minimum clear aperture for no vignetting:

$${\text{CA}}_{\min }=|y_a|+|yb|$$

Aspheric sag equation:

$$sag=z\left(r\right)=\frac{c{r}^2}{1+\sqrt{1-\left(\kappa +1\right){\left(cr\right)}^2}}+d{r}^4+e{r}^6+f{r}^8+g{r}^{10}+\dots$$

Gradient index profiles:

$$\begin{array}{l}n\left(r\right)=N_{00}+N_{10}{r}^2+N_{20}{r}^4+\dots \\ n\left(z\right)=N_{00}+N_{01}z+N_{02}{z}^2+\dots \end{array}$$

Merit function:

$$\text{ϕ}=\begin{array}{c}m\\ \\ i-1\end{array}{w}_i^2{\left(c_i-t_i\right)}^2$$

Athermalization condition:

$$\frac{{\mathrm{df}}_{\mathrm{Lens}}}{dT}=CT{E}_1{d}_{1}CT{E}_2{d}_2$$

Bireflectance scattering distribution function:

$$BSDF({\text{θ}}_i,{\text{ϕ}}_i;{\text{θ}}_o,{\text{ϕ}}_o)=\left(\frac{L}{E}\right)s{r}^{-1}$$

Sellmeier dispersion:

$$n\left(\text{λ}\right)=\sqrt{1+\frac{c_1{\text{λ}}^2}{{\text{λ}}^2-c_4}+\frac{c_2{\text{λ}}^2}{{\text{λ}}^2-c_5}+\frac{c_3{\text{λ}}^2}{{\text{λ}}^2-c_6}}$$

Schott dispersion:

$$n\left(\text{λ}\right)=\sqrt{c_1+c_2{\text{λ}}^2+\frac{c_3}{{\text{λ}}^2}+\frac{c_4}{{\text{λ}}^4}+\frac{c_5}{{\text{λ}}^6}+\frac{c_6}{c^8}}$$

Snell’s law:

$$n\sin \text{θ}=n\prime \sin \text{θ}\prime$$

Paraxial ray tracing:

$$\begin{array}{l}n\prime u\prime =nu-y\text{ϕ}\\ y\prime =y+n\prime u\prime \left(\frac{d}{n\prime }\right)\\ \text{ϕ}=c\left(n\prime -n\right)\end{array}$$

Lens maker’s equation and linear magnification:

$$\frac{1}{s\prime }=\frac{1}{f}+\frac{1}{s}m=\frac{h\prime }{h}=\frac{f}{s+f}$$

Thin lens power:

$$\text{Φ}=\frac{1}{f}=\left(c_1-c_2\right)\left(n-1\right)$$

Diffraction gratings:

$$m\text{λ}=d\left[\sin \left({\text{θ}}_m\right)-\sin \left({\text{θ}}_i\right)\right]{\text{λ}}_{blaze}=2d\sin \alpha$$

Sampling ratio:

$$Q=\frac{\text{λ}\left(f/\#\right)}{pixelpitch}$$

Lagrange invariant and étendue:

$$\begin{array}{l}H=n\overline{u}y-nu\overline{y}{n}^2\text{A}\text{Ω}={\pi }^2{H}^2\\ Etendue={\displaystyle \underset{surface}{\iint }d\text{A}a\text{Ω}}\end{array}$$