Lenticular optics

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Making a lenticular product requires not only a stringency in work and substantial precision throughout the manufacturing process but also a sound knowledge of lenticular optics. This article summarizes some of the important calculations of lenticular optics.

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[edit] 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.

[edit] Angle within the lens

The diagram at right shows in red 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.

[edit] Definitions

  • R is the angle between the extreme ray and the normal at the point where it exits the lens,
  • p is the pitch, or width of each lenticular cell,
  • r is the radius of curvature of the lenticule,
  • e is the thickness of the lenticular lens, and
  • n is the lens's index of refraction.

[edit] Calculation

R=A-\arctan\left({p \over h}\right),

where

A=\arcsin \left({p \over 2r}\right),
h = e − f is the distance from the back of the grating to the edge of the lenticule, and
f=r-\sqrt{r^2-\left({p \over 2}\right)^2}.

[edit] Angle outside the lens

The angle outside the lens is given by refraction of the ray determined above. The full angle of observation O is given by

O = 2(A − I),

where I is the angle between the extreme ray and the normal outside the lens. From Snell's Law,

I=\arcsin \left({n\sin(R) \over n_a}\right) ,

where n_a \approx 1.003 is the index of refraction of air.

[edit] 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 O would be 64.6°.

[edit] Focal distance of a lenticular network

One would expect that the focal plane of a lenticular lens should be designed to coincide with the back of the lens. The focal length of the lens is calculated from the lensmaker's equation, which in this case simplifies to:

F={r \over n-1},

where F is the focal length of the lens.

To find the rear focal plane, one needs to calculate the back focal distance, which is the distance from the rear of the lens to the focal plane. In this simple case, it is just

BFD=F- {e \over n}.

[edit] 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.

[edit] References