# 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 ${\displaystyle \Delta \lambda \simeq {\frac {\lambda d}{T\tan {\phi }}}\simeq {\frac {\lambda \pi \Delta n}{8n}}\simeq \lambda \arcsin {\left({\frac {1-Q}{1+Q}}\right)}}$ ${\displaystyle a_{n}=\arcsin {\left({\frac {\sin {a}}{n}}\right)}}$ ${\displaystyle b_{n}=\arcsin {\left({\frac {\sin {b}}{n}}\right)}}$ ${\displaystyle B={\frac {b_{n}-a_{n}}{2}}}$ ${\displaystyle {\text{Bragg ratio}}\beta ={\frac {T\lambda }{d^{2}}}}$ ${\displaystyle \phi \simeq \arcsin {n\sin {\left({\frac {a_{n}+b_{n}}{2}}\right)}}}$ ${\displaystyle {\text{Number of superimposed recordings}}\simeq {\frac {nT}{\lambda }}}$ ${\displaystyle {\text{Resolving Power}}{\frac {\lambda }{\Delta \lambda }}\simeq {\text{number of fringes}}}$

 Grating equation, reflection ${\displaystyle \displaystyle f_{0}\lambda =n(\cos {a}+\cos {b})}$ ${\displaystyle D.E.\simeq \tanh ^{2}{\left(\Delta nT/\left(\lambda \cos {\phi }\right)\right)}<99.9998\%}$

Uniform tilted reflector, also has weak transmission grating at surface.

 ${\displaystyle \Delta \lambda \simeq {\frac {\lambda d}{T}}}$ ${\displaystyle \Delta \theta \simeq {\frac {d}{T}}}$ ${\displaystyle \displaystyle 0<\Delta n<0.27}$ ${\displaystyle \displaystyle 3u ${\displaystyle \rho =Q{\frac {\lambda ^{2}}{d^{2}n\Delta n}}\simeq {\frac {2\pi \lambda T}{d^{2}n}}\simeq {\frac {2T}{d^{2}}}}$ ${\displaystyle {\frac {1}{\rho ^{2}}}\propto {\text{power lost to higher orders}}}$