Projecting 3D Image

From the post "Phase Shift, No Normal" dated 07 Jan 2018,

$\angle P=\cfrac{cf+cf' }{\lambda_i}{sin(\theta_i)}*\pi$

where

$cf'=\lambda _{ r }\left[ tan(\theta _{ r })-{ 2 }{ cosec(2\theta _{ r }) }+{ cosec(\theta _{ r }) } \right]$

and

$cf=\lambda _{ i }\left[ tan(\theta _{ i })-{ 2 }{ cosec(2\theta _{ i }) }+{ cosec(\theta _{ i }) } \right]$

Specifically, if we consider,

$\cfrac { \lambda _{ i } }{ \lambda _{ r } } =\cfrac { sin(\theta _{ i }) }{ sin(\theta _{ r }) } =1.5$

$\theta _{ r }=sin^{-1}\left(\cfrac { sin(\theta _{ i }) }{ 1.5 }\right)$

and plot $\angle P$ with respect to $\theta_i$ only, we have,

When we zoomed in $-\small{\cfrac{\pi}{2}}\le \theta_i\le\small{\cfrac{\pi}{2}}$,

$cf'$ is sinusoidal.  The phase different changes from zero up to $\approx 1.25\pi$.

Being able to quantify phase here, allows for manipulating phase in optical signals easier.

By varying phase, it is possible to project a pseudo-3D image out from a flat screen.