Four-region catadioptric tile

1. A four-region catadioptric tile, comprising:

a first surface;

a second surface; and

a filler material layer between said first and second surfaces,

wherein said four-region catadioptric tile has a first region wherein said first surface is convex and said second surface is convex, a second region wherein said first surface is convex and said second surface is concave, a third region wherein said first surface is concave and said second surface is convex, and a fourth region wherein said first surface is concave and said second surface is concave.

2. The four-region catadioptric tile of claim 1 , wherein said first surface comprises a transparent refraction surface and said second surface comprises a reflection surface.

3. The four-region catadioptric tile of claim 1 , wherein said first surface comprises a transparent refraction surface and said second surface comprises a transparent refraction surface.

4. The four-region catadioptric tile of claim 1 , wherein said four-region catadioptric tile is rectangular.

5. The four-region catadioptric tile of claim 1 , wherein said first surface is defined by a first equation:

Z e ={[(0.5 Y 0 /Tanφ)^2−(0.5( Y−Y 0 ))^2]^0.5}+{Σ i β i Sin(π f i X/X 0 )+ K}

wherein:

Z e is a thickness of said filler material layer between a reference plane and said first surface;

Y 0 is one half of a dimension of said four-region catadioptric tile measured along a first axis;

Y is a position relative to a center of said four-region catadioptric tile along said first axis;

φ is one half of an approximate total view-field angle along said first axis;

β i is amplitude of a maximum deflection relative to a mean value along a second axis;

f i is spatial frequency as a multiple of a frequency associated with 2X 0 frequency;

X 0 is one half of a dimension of said four-region catadioptric tile along said second axis;

X is a position relative to a center of said four-region catadioptric tile along said second axis; and

K is an offset that keeps said first and second surfaces from physically interfering.

6. The four-region catadioptric tile of claim 5 , wherein said second surface is defined by a second equation:

Z r ={[( X 0 /2Sinθ)^2−( X− 0.5 X 0 )^2]^0.5+[0.5 X 0 /Tanθ)]}

+{0.5 X 0 [(1/Sinθ)−(1/Tanθ)]}+{Σ j α j Sin(πμ j Y/Y 0 )+ K}

wherein:

Z r is a thickness of said filler material layer between said reference plane and said second surface;

θ is one quarter of an approximate total view-field angle along said second axis;

α j is amplitude of a maximum deflection relative to a mean value along said first axis; and

μ j is spatial frequency as a multiple of a frequency associated with the 2Y 0 frequency.

7. The four-region catadioptric tile of claim 6 , wherein a maximum value for the sum of Z e plus Z r is 5 mm or less, wherein X 0 is equal to or less than 5 mm, wherein Y 0 is equal to or less than 5 mm, wherein α j and β i equal to or less than 1.0, and wherein f i and μ j equal to or less than 1.0.

8. The four-region catadioptric tile of claim 6 , wherein a maximum value for the sum of Z e plus Z r is 1 mm or less, wherein X 0 is equal to or less than 1 mm, wherein Y 0 is equal to or less than 1 mm, wherein α j and β i equal to or less than 0.2, and wherein f i and μ j equal to or less than 0.2.

9. The four-region catadioptric tile of claim 6 , wherein a maximum value for the sum of Z e plus Z r is 0.5 mm or less, wherein X 0 equal to or less than 0.5 mm, wherein Y 0 equal to or less than 0.5 mm, wherein α j and β i equal to or less than 0.1, and wherein f i and μ j equal to or less than 0.1.

10. The four-region catadioptric tile of claim 6 , wherein a maximum value for the sum of Z e plus Z r is 0.25 mm or less, wherein X 0 is equal to or less than 0.25 mm, and wherein Y 0 is equal to or less than 0.25 mm.

11. The four-region catadioptric tile of claim 6 , further comprising a thin layer disposed on said first surface, said thin layer having an index of refraction;

wherein X 0 , θ, Y 0 , φ and said index of refraction of said thin layer are selected such that the portion of light that enters said four-region catadioptric tile from an off-axis angle equal to or greater than 60 degrees is on average captured within said filler material layer for a minimum of three times the mean thickness of said four-region catadioptric tile.

12. The four-region catadioptric tile of claim 6 , further comprising a thin layer disposed on said first surface, said thin layer having an index of refraction;

wherein X 0 , θ, Y 0 , φ and said index of refraction of said thin layer are selected such that the portion of light that enters said four-region catadioptric tile from an off-axis angle equal to or greater than 60 degrees is on average captured within said filler material layer for a minimum of five times the mean thickness of said four-region catadioptric tile.

13. The four-region catadioptric tile of claim 6 , further comprising a thin layer disposed on said first surface, said thin layer having an index of refraction;

wherein X, θ, Y 0 , φ and said index of refraction of said thin layer are selected such that the portion of light that enters said four-region catadioptric tile from an off-axis angle equal to or greater than 60 degrees is on average captured within said filler material layer for a minimum of ten times the mean thickness of said four-region catadioptric tile.

14. The four-region catadioptric tile of claim 5 , β i , and f i are selected such that the portion of light that is reflected at said first surface is reduced in brightness as if spread by at least five times the width of said four-region catadioptric tile.

15. The four-region catadioptric tile of claim 5 , β i , and f i are selected such that the portion of light that is reflected at said first surface is reduced in brightness as if spread by at least ten times the width of said four-region catadioptric tile.

16. The four-region catadioptric tile of claim 5 , β i , and f i are selected such that the portion of light that is reflected at said first surface is reduced in brightness as if spread by at least twenty times the width of said four-region catadioptric tile.

17. The four-region catadioptric tile of claim 6 , wherein said first surface's axis comprises a vertical axis and said second surface's axis comprises a horizontal axis.

18. The four-region catadioptric tile of claim 1 , further comprising a thin layer disposed on said first surface.

19. The four-region catadioptric tile of claim 18 , wherein said thin layer has a refractive index higher than a refractive index of said filler material layer.

20. The four-region catadioptric tile of claim 18 , wherein said thin layer has a refractive index at least 10% higher than a refractive index of said filler material layer.

21. The four-region catadioptric tile of claim 18 , wherein said thin layer has a refractive index at least 25% higher than a refractive index of said filler material layer.

22. The four-region catadioptric tile of claim 18 , wherein said thin layer has a refractive index at least 50% higher than a refractive index of said filler material layer.

23. The four-region catadioptric tile of claim 1 , wherein said filler material layer comprises an absorptive element.

24. The four-region catadioptric tile of claim 1 , wherein said filler material layer comprises multiple layers.

25. The four-region catadioptric tile of claim 1 , wherein said filler material layer has a refractive taper.

26. The four-region catadioptric tile of claim 1 , wherein said four-region catadioptric tile is prepared by any process selected from the group comprising machining, milling, etching, photo-process, broaching, roll forming, flat pressing, stamping, vacuum forming, electroforming, film suspension, molding, casting, injection, deposition, printing, and precipitation.

27. A tessellated projection screen comprising:

a plurality of catadioptric tiles, each catadioptric tile having four regions;

each of said catadioptric tiles comprising:

a first surface;

a second surface; and

a filler material layer between said first and second surfaces,

wherein said four-region catadioptric tile has a first region wherein said first surface is convex and said second surface is convex, a second region wherein said first surface is convex and said second surface is concave, a third region wherein said first surface is concave and said second surface is convex, and a fourth region wherein said first surface is concave and said second surface is concave.

28. The tessellated projection screen of claim 27 , wherein said first surface is defined by a first equation:

Z e ={[(0.5 Y 0 /Tanφ)^2−(0.5( Y−Y 0 ))^2]^0.5}+{Σ i β i Sin(π f i X/X 0 )+ K}

wherein:

Z e is a thickness of said filler material layer between a reference plane and said first surface;

Y 0 is one half of a dimension of said four-region catadioptric tile measured along a first axis;

Y is a position relative to a center of said four-region catadioptric tile along said first axis;

φ is one half of an approximate total view-field angle along said first axis;

β i is amplitude of a maximum deflection relative to a mean value along a second axis;

f i is spatial frequency as a multiple of a frequency associated with 2X 0 frequency;

X 0 is one half of a dimension of said four-region catadioptric tile along said second axis;

X is a position relative to a center of said four-region catadioptric tile along said second axis; and

K is an offset that keeps the entry and reflective surfaces from physically interfering.

29. The tessellated projection screen of claim 27 , wherein said second surface is defined by a second equation:

Z r ={[( X 0 /2Sinθ)^2−( X− 0.5 X 0 )^2]^0.5+[0.5 X 0 /Tanθ)]}

+{0.5 X 0 [(1/Sinθ)−(1/Tanθ)]}+{Σ j α j Sin(πμ j Y/Y 0 )+ K}

wherein:

Z r is a thickness of said filler material layer between said reference plane and said second surface;

θ is one quarter of an approximate total view-field angle along said second axis;

α j is amplitude of a maximum deflection relative to a mean value along said first axis; and

μ j is spatial frequency as a multiple of a frequency associated with the 2Y 0 frequency.

30. An angularly dependent retroreflector comprising:

a plurality of catadioptric tiles, each catadioptric tile having four regions;

each of said catadioptric tiles comprising:

a first surface;

a second surface; and

a filler material layer between said first and second surfaces,

wherein said four-region catadioptric tile has a first region wherein said first surface is convex and said second surface is convex, a second region wherein said first surface is convex and said second surface is concave, a third region wherein said first surface is concave and said second surface is convex, and a fourth region wherein said first surface is concave and said second surface is concave.