# Equations

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## Common DOE surfaces

### Sinusoidal, triangle

 Max D.E. ~ 30% ± 1 order Power distribution follows Bessel function in the scaler region. T ≈ λ for transmission. T ≈ λ/4 for reflection. Easily replicated or deep etch can yield higher efficiency.

### Square, rectangle

 Max D.E. ~ 40% ± 1 order. Power distributions favors odd orders. Pi phase shift typical. Classic binary grating. T ≈ λ for transmission. T ≈ λ/4 for reflection.

### Blazed, sawtooth

 Max D.E ~ 98% Reflection Case: λ = 2 d sin a T = λ/2 ± 10% Transmission Case: λ = d sin b a ≈ 2b T = λ/(n-1) ± 10% Surface shadows reduce D.E. by ${\displaystyle \textstyle {\frac {\lambda ^{2}}{nd^{2}}}}$ Wavelength changes reduce D.E. by ${\displaystyle \textstyle \left({\frac {\pi \Delta \lambda }{b\lambda }}\right)^{2}}$

## Holographic Deflectors (hologons)

 N = θD/(1.4λ) N = number of resolvable spots θ = full scan angle in radians D = beam diameter in microns λ = wavelength Φ = wedge angle of glass Special case of linear scan and near wobble invariance a ~ b ~ 45° max straight scan angle ~ 36° Dynamic wobble ${\displaystyle \scriptstyle \simeq \Delta \Omega /128}$ Wedge wobble = (n-1)Φ Scan angle multiplier = 1.4 @ 90° cross scan angle

Last modified on 9/29/97