Abstract

Lenticular transmission and reflection projection screens are considered with a view to explaining some experimental effects that cannot be accounted for by geometrical optics. When illuminated with white light, the screens display color bands and, at the edges of the patterns, white and dark lines appear. These phenomena can be explained by diffraction theory and analysis by the method of stationary phase.

© 1975 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford and New York, 1970), Appendix III.
  2. Discussions relating to various unwanted features observed experimentally in certain types of projection screens, as well as further references in this area can be found in the following patents: (1) U. S. No. 3, 754, 811, Aug. 28, 1973; (2) U.S. No. 3, 754, 813, Aug. 28, 1973; and (3) U. S. No. 3, 788, 171, Jan. 29, 1974.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford and New York, 1970), Appendix III.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford and New York, 1970), Appendix III.

Other (2)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford and New York, 1970), Appendix III.

Discussions relating to various unwanted features observed experimentally in certain types of projection screens, as well as further references in this area can be found in the following patents: (1) U. S. No. 3, 754, 811, Aug. 28, 1973; (2) U.S. No. 3, 754, 813, Aug. 28, 1973; and (3) U. S. No. 3, 788, 171, Jan. 29, 1974.

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Figures (6)

FIG. 1
FIG. 1

Single concave lenticule.

FIG. 2
FIG. 2

Illuminance as a function of angle.

FIG. 3
FIG. 3

Multiple convex lenticules.

FIG. 4
FIG. 4

Convex lenticular screen, illuminance based on combined first- and second-order contributions. p = 1.5717 mm 1, d = 0.2034 mm.

FIG. 5
FIG. 5

Pair of alternate convex and concave lenticules.

FIG. 6
FIG. 6

Alternating screen, illuminance based on first-order contributions from the two lenticules, p = 1.5717 mm 1, d = 0.2034 mm.

Equations (51)

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P 0 N = y tan θ a , P 0 M = P 0 N cos θ = y sin θ a cos θ , [ O 0 P 0 ] [ O P ] = n a a cos θ + y sin θ .
U 0 ( θ ) = d / 2 d / 2 [ 1 + ( d a / d y ) 2 ] 1 / 2 × exp { i k [ a ( n cos θ ) + y sin θ ] } d y .
I 0 ( θ ) = | U 0 ( θ ) | 2
a = p y 2 , d a / d y = 2 p y .
C ( k ) = A B G ( y ) exp [ i k F ( y ) ] d y
C 0 ( k ) = G ( y 0 ) [ 2 π / k | F ( y 0 ) | ] 1 / 2 × exp [ i k F ( y 0 ) ± i π / 4 ] ,
F ( y ) = p y 2 ( n cos θ ) + y sin θ , F ( y ) = 2 p y ( n cos θ ) + sin θ , F ( y ) = 2 p ( n cos θ ) , G ( y ) = [ 1 + ( 2 p y ) 2 ] 1 / 2 .
y 0 = sin θ / 2 p ( n cos θ ) , F ( y 0 ) = sin 2 θ / 4 p ( n cos θ ) , F ( y 0 ) = 2 p ( n cos θ ) > 0 , G ( y 0 ) = ( 1 + n 2 2 n cos θ ) 1 / 2 / ( n cos θ ) .
U 0 ( θ ) = [ ( λ / 2 p ) ( 1 + n 2 2 n cos θ ) ] 1 / 2 ( n cos θ ) 3 / 2 × exp [ i k F ( y 0 ) + i π / 4 ] ,
I 0 ( θ ) = ( λ / 2 p ) ( 1 + n 2 2 n cos θ ) ( n cos θ ) 3 .
| sin θ | < p d ( n cos θ ) .
sin θ = p d ( n cos θ ) .
U ( θ ) = j = 0 N 1 d / 2 d / 2 [ 1 + ( d a / d y ) 2 ] 1 / 2 × exp { i k [ a ( n cos θ ) + ( y + j d ) sin θ ] } d y = U 0 ( θ ) P ( θ ) ,
P ( θ ) = j = 0 N 1 e i j k d sin θ .
I ( θ ) = I 0 ( θ ) Q ( θ ) , Q ( θ ) = | P ( θ ) | 2 .
Q ( θ ) = | 1 exp ( i N D ) 1 exp ( i D ) | 2 = 1 cos ( N D ) 1 cos D = sin 2 ( N D / 2 ) sin 2 ( D / 2 ) ,
D = k d sin θ , k = 2 π / λ .
D / 2 = ( k d / 2 ) sin θ = m π
sin θ = m λ / d ,
Δ θ λ / d cos θ
C ( k ) = A B [ G ( y ) / i k F ( y ) ] i k F ( y ) exp [ i k F ( y ) ] d y
C ( k ) = [ G ( y ) / i k F ( y ) ] exp [ i k F ( y ) ] | B A + ( 1 / i k ) A B [ G ( y ) / F ( y ) ] exp [ i k F ( y ) ] d y .
C ( k ) C 1 / i k + C 2 / ( i k ) 2 + .
C ( k ) [ G ( y ) / i k F ( y ) ] exp [ i k F ( y ) ] | B A .
U ¯ 0 ( θ ) = ( 1 / i k ) exp ( i E ) × { [ p d ( n cos θ ) + sin θ ] 1 exp ( i D / 2 ) + [ p d ( n cos θ ) sin θ ] 1 exp ( i D / 2 ) ] } ,
D = k d sin θ , k = 2 π / λ
E = ( k p d 2 / 4 ) ( n cos θ ) .
U ¯ 0 ( θ ) = ( 2 / i k ) exp ( i E ) × [ ( p d ) 2 / ( n cos θ ) 2 sin 2 θ ] 1 × [ p d ( n cos θ ) cos ( D / 2 ) i sin θ sin ( D / 2 ) ] .
I ¯ 0 ( θ ) = | U ¯ 0 ( θ ) | 2 = ( 2 / k ) 2 { [ p d ( n cos θ ) cos ( D / 2 ) ] 2 + sin 2 θ × sin 2 ( D / 2 ) } [ ( p d ) 2 ( n cos θ ) 2 sin 2 θ ] 2 .
I 0 ( ± θ 1 ) = ( λ / 8 p ) ( 1 + n 2 2 n cos θ 1 ) ( n cos θ 1 ) 3 ,
U ˆ 0 ( θ ) = U 0 ( θ ) + U ¯ 0 ( θ ) , I ˆ 0 ( θ ) = | U ˆ 0 ( θ ) | 2 ,
U 0 ( θ ) = d / 2 d / 2 G ( y ) exp [ i k ( F y ) ] d y ,
F ( y ) = p y 2 ( 1 + cos θ ) y sin θ , G ( y ) = [ 1 + ( 2 p y ) 2 ] 1 / 2 .
U 0 ( θ ) ( λ / p ) 1 / 2 ( 1 + cos θ ) 1 exp ( i A ) ,
A = ( π / 2 ) [ ( cos θ 1 ) / λ p + 1 2 ] ,
I 0 ( θ ) = ( λ / p ) ( 1 + cos θ ) 2 .
θ 1 = 2 tan 1 ( p d ) ,
U ¯ 0 ( θ ) = ( 1 / i k ) [ 1 + ( p d ) 2 ] 1 / 2 exp ( i k b d / 4 ) × [ ( b + sin θ ) 1 exp ( i D / 2 ) + ( b sin θ ) 1 exp ( i D / 2 ) ] ,
b = p d ( 1 + cos θ ) , D = k d sin θ .
U ¯ 0 ( θ ) = ( 1 ) m ( λ b / i π ) [ 1 + ( p d ) 2 ] 1 / 2 exp ( i k b d / 4 ) .
I ˆ ( θ ) = | U 0 ( θ ) + U ¯ 0 ( θ ) | 2
I ¯ ( θ ) = | U ¯ 0 ( θ ) | 2
V 0 ( θ ) = d / 2 d / 2 [ 1 + ( 2 p y ) 2 ] 1 / 2 × exp { i k [ ( 2 c a ) ( 1 + cos θ ) + ( d y ) sin θ ] } d y ,
a = p y 2 , c = p d 2 / 4 .
V 0 ( θ ) ( λ / p ) exp ( i B ) ,
B = D π / 4 + ( π / λ ) [ b d + ( 1 cos θ ) / 2 p ] ,
V ¯ 0 ( θ ) = ( 1 / i k ) [ 1 + ( p d ) 2 ] 1 / 2 exp ( i k b d / 4 ) × [ ( b + sin θ ) 1 exp ( i D / 2 ) ] ( b sin θ ) 1 exp ( 3 i D / 2 ) .
U ¯ 0 ( θ ) + V ¯ 0 ( θ ) = ( 1 / i k ) [ 1 + ( p d ) 2 ] 1 / 2 exp ( i k b d / 4 ) × ( b sin θ ) 1 [ exp ( i D / 2 ) exp ( 3 i D / 2 ) ] ,
[ exp ( i D / 2 ) exp ( 3 i D / 2 ) ] = 2 i sin D exp ( i D / 2 ) ,
U 0 ( θ ) + V 0 ( θ ) = ( λ / p ) [ ( 1 + cos θ ) 1 exp ( i A ) + exp ( i B ) ] ,
I ( θ ) = | U 0 ( θ ) + V 0 ( θ ) | 2 ,