VHOE Relationships

VHOE Relationships

Key
a = input angle in air	b = output angle in air	a_n = input angle in medium
b_n = output angle in medium	f₀ = spatial frequency = 1/d	$Lambda$ = wavelength
n = index of refraction	$Delta$ n = index modulation	D.E. = diffraction efficiency
$Phi$ = The half angle	d = grating period	T = thickness of medium
$Rho$ = 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
f₀ $Lambda$ = sin a + sin b
D.E. $~$ sin² [ $Delta$ n T / ( $Lambda$ cos $Phi$ ) ] < 99.9%

Plane grating, slanted fringes, +3 order is TIR, $Delta$ n is asymmetric

a_n = arcsin [ (sin a) / n ]
b_n = arcsin [ (sin b) / n ]
B = (b_n - a_n)/2
Bragg ratio $Beta$ = T $Lambda$ /d²
$Phi$ $~$ arcsin ( n sin [ (a_n + b_n)/2 ] )
Number of superimposed recordings $~$ $Delta$ n T/ $Lambda$
Resolving Power $Lambda$ / $Delta$ $Lambda$ $~$ N = # of fringes

Grating equation, reflection
f₀ $Lambda$ = n (cos a + cos b)
D.E. $~$ tanh² [ $Delta$ n T / ( $Lambda$ cos $Phi$ ) ] < 99.9998%

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

0 < $Delta$ n < 0.27
3u < T < 50u
power lost to higher orders

Last modified on 9/29/97
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