Angle of view of a lenticular print

Update: 2020-11-27 16:11      View:

Angle of view of a lenticular print

The angle of view of a lenticular print is the range of angles within which the observer can see the entire image. This is determined by the maximum angle at which a ray can leave the image through the correct lenticule.

Angle within the lens

 

Sq3d-angle-refraction.gif
The diagram at right shows in green the most extreme ray within the lenticular lens that will be refracted correctly by the lens. This ray leaves one edge of an image strip (at the lower right) and exits through the opposite edge of the corresponding lenticule.

Definitions

 

  •  is the angle between the extreme ray and the normal at the point where it exits the lens,
  •  is the pitch, or width of each lenticular cell,
  •  is the radius of curvature of the lenticule,
  •  is the thickness of the lenticular lens
  •  is the thickness of the substrate below the curved surface of the lens, and
  •  is the lens's index of refraction.

Calculation

 

  • ,
where
  • ,
  •  is the distance from the back of the grating to the edge of the lenticule, and
  • .

Angle outside the lens

 

Lr3d-angle-indice.gif
The angle outside the lens is given by refraction of the ray determined above. The full angle of observation  is given by
  • ,
where  is the angle between the extreme ray and the normal outside the lens. From Snell's Law,
  • ,
where  is the index of refraction of air.

Example

Consider a lenticular print that has lenses with 336.65 µm pitch, 190.5 µm radius of curvature, 457 µm thickness, and an index of refraction of 1.557. The full angle of observation  would be 64.6°.

Rear focal plane of a lenticular network

The focal length of the lens is calculated from the lensmaker's equation, which in this case simplifies to:


  • ,
where  is the focal length of the lens.
The back focal plane is located at a distance  from the back of the lens:
A negative BFD indicates that the focal plane lies inside the lens.
In most cases, lenticular lenses are designed to have the rear focal plane coincide with the back plane of the lens. The condition for this coincidence is , or
This equation imposes a relation between the lens thickness  and its radius of curvature .

Example

The lenticular lens in the example above has focal length 342 µm and back focal distance 48 µm, indicating that the focal plane of the lens falls 48 micrometers behind the image printed on the back of the lens.