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# Equation Summary

Numerical aperture and f /# :

$NA=n\prime \mathrm{sin}u\prime \text{ }\text{\hspace{0.17em}}f/#=\frac{f}{EPD}$

Rayleigh criterion:

$\text{Δ}X=\frac{0.61\text{λ}}{NA}$

Image height as a function of field angle:

$h\prime =f\mathrm{tan}\text{θ}$

Transverse ray error:

${\epsilon }_{y}^{\prime }=\frac{1}{n\prime u{\prime }_{a}}\frac{\partial W}{\partial {\text{ρ}}_{y}}\text{\hspace{0.17em}}{\epsilon }_{x}^{\prime }=\frac{1}{n\prime u{\prime }_{a}}\frac{\partial W}{\partial {\text{ρ}}_{x}}$

Strehl ratio:

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

Wavefront aberration polynomial:

$\begin{array}{l}{W}_{IJK}⇒{H}^{I}{\text{ρ}}^{J}{\mathrm{cos}}^{K}\text{θ}\\ W\left(H,\text{ρ},\text{θ}\right)={W}_{020}{\text{ρ}}^{2}+{W}_{111}H\text{ρ}\mathrm{cos}\text{θ}+{W}_{040}{\text{ρ}}^{4}+{W}_{131}H{\text{ρ}}^{3}\mathrm{cos}\text{θ}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{W}_{222}{H}^{2}{\text{ρ}}^{2}{\mathrm{cos}}^{2}\text{θ}+{W}_{220}{H}^{2}{\text{ρ}}^{2}+{W}_{311}{H}^{3}\text{ρ}\mathrm{cos}\text{θ}+O\left(6\right)\end{array}$

Contrast:

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

Focal lengths of any two thin lens system:

${f}_{a}=\frac{df}{f-BFL}\text{ }\text{\hspace{0.17em}}{f}_{b}=\frac{dBFL}{f-BFL-d}$

Zero-Petzval solution for two thin lenses:

${f}_{a}=-{f}_{b}=f-BFL\text{\hspace{0.17em}}\text{ }d=\frac{{\left(f-BFL\right)}^{2}}{f}$

Two-mirror solution:

${c}_{1}=\frac{BFL-f}{2df}\text{\hspace{0.17em}}\text{ }{c}_{2}=\frac{BFL+d-f}{2dBFL}$

Schwarzchild solution:

$d=2f\text{\hspace{0.17em}}\text{ }{c}_{1}=\left(\sqrt{5}-1\right)f\text{ }c2=\left(\sqrt{5}+1\right)f$

Aplanatic condition:

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

Bending and shape factors:

$\text{β}=\frac{{c}_{1}+{c}_{2}}{{c}_{1}-{c}_{2}}\text{ }C=\frac{{u}_{a}+{u}_{a}^{\prime }}{{u}_{a}-{u}_{a}^{\prime }}$

Thin lens bending for minimum spherical:

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

Thin lens bending for minimum coma:

$\frac{{c}_{2}}{{c}_{1}}=\frac{\left({n}^{2}-n-1\right)}{{n}^{2}}$

Achromatic doublet:

$\text{Φ}={\text{ϕ}}_{1}+{\text{ϕ}}_{2}\text{ }{\text{ϕ}}_{1}=\text{Φ}\frac{{V}_{1}}{{V}_{1}-{V}_{2}}\text{ }{\text{ϕ}}_{2}=-\text{Φ}\frac{{V}_{2}}{{V}_{1}-{V}_{2}}$

Petzval sum:

$\sum _{j}\frac{{\text{ϕ}}_{j}}{{n}_{j}}$

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:

${\text{CA}}_{\mathrm{min}}=|{y}_{a}|+|yb|$

Aspheric sag equation:

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

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

Merit function:

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

Athermalization condition:

$\frac{{\mathrm{df}}_{\mathrm{Lens}}}{dT}=CT{E}_{1}{d}_{1}\text{ }CT{E}_{2}{d}_{2}$

Bireflectance scattering distribution function:

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

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:

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

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:

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

Thin lens power:

$\text{Φ}=\frac{1}{f}=\left({c}_{1}-{c}_{2}\right)\left(n-1\right)$

Diffraction gratings:

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

Sampling ratio:

$Q=\frac{\text{λ}\left(f/#\right)}{pixel\text{\hspace{0.17em}}pitch}$

Lagrange invariant and étendue:

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

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