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Biconvex, Thin, Paraxial-Ray Focal Lens

Biconvex, Thin, Paraxial-Ray Focal Lens - Biconvex Lens...

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Unformatted text preview: Biconvex Lens - Thin-Lens Paraxial-Ray Focal Length A biconvex lens is used to focus a plane wave that is propagating on-axis (light traveling parallel to the axis of lens). The lens has a diameter D, a center thickness d, and an index of refraction of n. The magnitudes of the radii of curvature of the surfaces are R1 and R2. This lens and this configuration are shown in the attached figure. Starting from the exact expression for the focal length given with the figure, cal— culate, showing all work, the expression for the thin-lens paraxial-ray focal length for this lens. In the paraxial ray approximation, all rays make small angles with respect to the axis of the optical system. Further, for a thin lens, the center thickness of the lens d —+ 0. Simplify your answer as much as possible. Put your final answer in the space , provided. For a biconvex lens that has D = 32 mm, d = 5.26 mm, n = 1.5, and the mag— nitudes of the radii of curvature of the surfaces are R1 = 50 mm and R2 = 50 mm, numerically evaluate the thin—lens paraxial-ray focal length from your above expression. Express your answer in mm accurately to four significant figures. Put your answer in the space provided. Thin-lens paraxial-ray focal length (analytic expression) Thin—lens paraxial-ray focal length (numerical value) 2 mm BICONVEX LENS A biconvex lens has radii of curvature of R1 and R2, a center thickness of d, and an index of refraction n. If R1 and R2 are taken to be magnitudes, the focal distance of this biconvex lens in air for an axial ray of distance h from the principal axis is f = {n{(R1+ R2 — d) swim—1 (R31) ’ Sin—1 (5%)] + 3/ sin[(m-1{ (R12) + R2 _ d) W (32:) — (”—291 + 1%» _(sm-l{ (1%le + R2 _ d) “am—1%) - WG-29] + z’é-zb + [ (7:2) — (fl } - R2 5mm Fig. 1. Focusing of rays by biconvex lens showing spherical aberration. Biconvex Lens - Thin-Lens Paraxial-Ray Focal Length The focal distance of this biconvex lens in air for an axial ray of distance h from the principal axis is W“Wyn—1(a)—sm-I<n—2:>1+%}/ W[({(R>(R1+ R2 _ d) man-1 (121) - (%)l + £3) _(sm—1{(;2)(Rl - d) m“ ‘15:) _ Shh—1%)] + ”’1; }) +ism—I<Ril>—m—l<n—an}—R2 The focal length for paraxial rays occurs for small angles and so sinG can be replaced by 6, etc. Also, for thin lenses, d ——> 0. Therefore {mm-+12» [(1:1) (n;1>]+h/ mam ms (%)]+%} .—‘{(;2><R1 R» [(122) (an «wan—R2 f ={n(R1+R2)h(:g11)+ h/ h(n— 1) + i ’an R2 [-gz—(Rl + R2) +h(n};1)]}—R2 f={h(n—1)(R1+R2) +h(n—1)R1/[h(n—1) (R1+R2)+ h(n—1) (n -- 1) R1 R1 R2 (n — 1)R2 _ (R1+R)(n—1)+R1 (R1(n——1)+nR2) R1 f_{ (Ti—”Bl /[ anRz +nR1R2]}-R2 _ [nR1+(n—1)R]R2 (R (n—1)2+n(n—1)R +(n—1)R1 f—{ (n—1)R1R: /[ 1 n(n—1)R'11;2 ]}—R2 f =[nR1 + (n —1)R2]R2/[(R1(n— 1)2 +n(n—- 1)R2 + (n— 1)R1] —R2 f _ n2 R1 R2 + n2 R3 — nRg — (n2R1 + n2R2 — nR1 — nR2)R2 _ (anl + n2R2 - an - nRz) R1 R2 R1 R2 1 f:(an+nR2—R1—R2) = (n—1)(R1+R2) : m which is equivalent to the thin lens focal length equation. For n = 1.5, R1 = 50 mm, and R2 = 50 mm, then f = 50.0 mm ...
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