# VHOE Relationships

 Key a = input angle in air b = output angle in air an = input angle in medium bn = output angle in medium f0 = spatial frequency = 1/d λ = wavelength n = index of refraction Δn = index modulation D.E. = diffraction efficiency φ = The half angle d = grating period T = thickness of medium ρ = regime factor Q = quality factor B = fringe tilt angle 0, +1, -1, +2, -2 = diffraction orders possible f = focal length f# = f number

 Grating equation, transmission f0λ = sin a + sin b D.E. ~ sin2 [Δn T / (λ cos φ)] < 99.9%

 Plane grating, slanted fringes, +3 order is TIR, Δn is asymmetric $\Delta \lambda \simeq {\frac {\lambda d}{T\tan {\phi }}}\simeq {\frac {\lambda \pi \Delta n}{8n}}\simeq \lambda \arcsin {\left({\frac {1-Q}{1+Q}}\right)}$ $a_{n}=\arcsin {\left({\frac {\sin {a}}{n}}\right)}$ $b_{n}=\arcsin {\left({\frac {\sin {b}}{n}}\right)}$ $B={\frac {b_{n}-a_{n}}{2}}$ ${\text{Bragg ratio}}\beta ={\frac {T\lambda }{d^{2}}}$ $\phi \simeq \arcsin {n\sin {\left({\frac {a_{n}+b_{n}}{2}}\right)}}$ ${\text{Number of superimposed recordings}}\simeq {\frac {nT}{\lambda }}$ ${\text{Resolving Power}}{\frac {\lambda }{\Delta \lambda }}\simeq {\text{number of fringes}}$ Grating equation, reflection $\displaystyle f_{0}\lambda =n(\cos {a}+\cos {b})$ $D.E.\simeq \tanh ^{2}{\left(\Delta nT/\left(\lambda \cos {\phi }\right)\right)}<99.9998\%$ Uniform tilted reflector, also has weak transmission grating at surface. $\Delta \lambda \simeq {\frac {\lambda d}{T}}$ $\Delta \theta \simeq {\frac {d}{T}}$ $\displaystyle 0<\Delta n<0.27$ $\displaystyle 3u $\rho =Q{\frac {\lambda ^{2}}{d^{2}n\Delta n}}\simeq {\frac {2\pi \lambda T}{d^{2}n}}\simeq {\frac {2T}{d^{2}}}$ ${\frac {1}{\rho ^{2}}}\propto {\text{power lost to higher orders}}$ 