Abstract

Mathematical expressions for the primary aberrations of a thin, meniscus Fresnel lens (grooved on both surfaces) are derived and compared to the corresponding expressions for the case of a thin, flat Fresnel lens. It is shown that the only aberrations that differ are spherical aberration and line coma, both of which are independent of stop position. Moreover, a meniscus Fresnel lens has an additional degree of freedom available for aberration correction. Some applications of the theory are discussed.

© 1976 Optical Society of America

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References

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  1. E. Delano, J. Opt. Soc. Am. 64, 459 (1974).
    [Crossref]
  2. Note that the values of ΔSl* and ΔS2l* given in Eqs. (11) are not to be confused with the values given in Eqs. (30) and (31) in I, which refer to the effect of figuring, and which is independent of the effect of using a meniscus.

1974 (1)

J. Opt. Soc. Am. (1)

Other (1)

Note that the values of ΔSl* and ΔS2l* given in Eqs. (11) are not to be confused with the values given in Eqs. (30) and (31) in I, which refer to the effect of figuring, and which is independent of the effect of using a meniscus.

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Figures (5)

FIG. 1
FIG. 1

Comparison of a thin, meniscus Fresenel lens (right) and a thin, flat Fresnel lens (center), with the conventional lens (left) from which they are derived. The width of the grooves is greatly exaggerated.

FIG. 2
FIG. 2

Bending a thin, meniscus Fresnel lens. The power of the lens and the curvature of the meniscus are kept fixed as power is shifted from the first surface of the lens to the second, in the sequence (a), (b), (c).

FIG. 3
FIG. 3

Side view (a) and top view (b) of a skew ray originating at the object point P which lies in the x, y plane at a height h above the x axis.

FIG. 4
FIG. 4

Side view (a) and top view (b) of the skew ray after refraction. The point P′ is the ideal image point corresponding to the object point P in Fig. 3. The principal ray is not shown.

FIG. 5
FIG. 5

Optical schematic of a 1:1 relay lens consisting of two identical meniscus Fresnel lenses a and b which are symmetrically placed about a central stop. The marginal ray (solid) and principal ray (dashed) are shown.

Equations (25)

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ν = - y 0 c + 1 2 c 3 y 0 ( y 0 2 + z 0 2 ) + O ( 5 ) , λ = - z 0 c + 1 2 c 3 z 0 ( y 0 2 + z 0 2 ) + O ( 5 ) ,
( n - 1 ) ( ν 1 - ν 2 ) = - K y 0 + 1 2 ( n - 1 ) ( c 1 3 - c 2 3 ) × y 0 ( y 0 2 + z 0 3 ) + O ( 5 ) , ( n - 1 ) ( λ 1 - λ 2 ) = - K z 0 + 1 2 ( n - 1 ) ( c 1 3 - c 2 3 ) × z 0 ( y 0 2 + z 0 2 ) + O ( 5 ) ,
y 0 = y + u x 0 = y + u x 0 , z 0 = z + v x 0 = z + v x 0 ,
y 0 = y + 1 2 u c 0 ( y 2 + z 2 ) + O ( 5 ) , z 0 = z + 1 2 v c 0 ( y 2 + z 2 ) + O ( 5 ) .
( n - 1 ) ( ν 1 - ν 2 ) = - K y - 1 2 K u c 0 ( y 2 + z 2 ) + 1 2 ( n - 1 ) ( c 1 3 - c 2 3 ) y ( y 2 + z 2 ) + O ( 5 ) , ( n - 1 ) ( λ 1 - λ 2 ) = - K z - 1 2 K v c 0 ( y 2 + z 2 ) + 1 2 ( n - 1 ) ( c 1 3 - c 2 3 ) z ( y 2 + z 2 ) + O ( 5 ) .
u = u - K y + δ u M = u - K y + δ u F - 1 2 K u c 0 ( y 2 + z 2 ) , v = v - K z + δ v M = v - K z + δ v F - 1 2 K v c 0 ( y 2 + z 2 ) ,
δ Y M = y + u l - h , δ Z M = z + v l .
y = y + 1 2 K c 0 y ( y 2 + z 2 ) + O ( 5 ) , z = z + 1 2 K c 0 z ( y 2 + z 2 ) + O ( 5 ) .
δ Y M / l = δ u F + 1 2 K c 0 ( y / l - u ) ( y 2 + z 2 ) + O ( 5 ) , δ Z M / l = δ v F + 1 2 K c 0 ( z / l - v ) ( y 2 + z 2 ) + O ( 5 ) .
y / l - u = ( ū - u ) - u = - 1 2 K y ( T - 1 ) - 1 2 K y ( T + 1 ) - ū ;
y / l - u = - ( K y T + ū ) , z / l - v = - K z T .
δ Y M / l = δ u F - 1 2 K 2 c 0 T y ( y 2 + z 2 ) - 1 2 K c 0 ū ( y 2 + z 2 ) + O ( 5 ) , δ Z M / l = δ v F - 1 2 K 2 c 0 T z ( y 2 + z 2 ) + O ( 5 ) .
δ Y M = δ Y F + Δ a 1 η ( η 2 + ζ 2 ) + Δ a 3 σ ( η 2 + ζ 3 ) + O ( 5 ) , δ Z M = δ Z F + Δ a 1 ζ ( η 2 + ζ 2 ) + O ( 5 ) ,
δ Y M = δ Y F + Δ b 1 ρ 3 cos θ + Δ b 2 l ρ 2 σ + O ( 5 ) , δ Z M = δ Z F + Δ b 1 ρ 3 sin θ + O ( 5 ) ,
Δ b 1 * = Δ b 1 ,             Δ b 2 l * = Δ b 2 l .
Δ S 1 = - 2 v m Δ b 1 = - 2 v m ( - 1 2 l K 2 r m 3 c 0 T ) ,
Δ S 1 * = Δ S 1 = - r m 4 K 2 c 0 T , Δ S 2 l * = Δ S 2 l = - Q m r m 2 K c 0 .
S 2 c = - 1 2 Q m r m 2 K 2 [ ( 1 + ω ) S + ( 2 + ω ) T ] = 0 , S 2 l = + 1 2 Q m r m 2 K 2 [ ( 1 + ω ) S + ω T ] - Q m r m 2 K c 0 = 0.
S 2 l = - Q m r m 2 K ( K T + c 0 ) = 0
K a = K b = 1 ,             A = 1 ,             B = 1 2 ,
u a = 1 ,             u a = 0 ,             u b = - 1 ,             y a = 1 ,             y b = 1 , ū a = 1 2 ,             ū a = 1 ,             ū b = 1 2 ,             y ¯ a - 1 2 ,             y ¯ b = 1 2 ,
S 1 a = S 1 b = - 1 4 y a 4 K a 3 [ γ a + 2 β a T a + ( 3 + ω ) T a 2 ] - y a 4 K a 2 ( c 0 ) a T a = 0 , S 3 a = S 3 b = - 1 4 y a 2 y ¯ a 2 K a 3 [ γ a + β a ( T a + T ¯ a ) + T ¯ a 2 + ( 2 + ω ) T a T ¯ a ] = 0 , S 4 a = S 4 b = - 1 2 Q y a y ¯ a K a 2 ( β a + ω T ¯ a ) = 0 ,
γ = ( ω - 1 ) S 2 + ( 1 - ω ) - 1 - 4 y - 4 K - 3 Δ S 1
β a = - β b = 1 ,             γ a = γ b = 9 4 ,             ( c 0 ) a = - ( c 0 ) b = - 95 48 .
S a = - S b = 3 5 ,             ( Δ S 1 ) a = ( Δ S 1 ) b = 63 400 .