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Designing an Achromatic Doublet, I Achromat The purpose is to give the same power for red and blue light. For touching two thin lens, we have the total power (red) $1,R + 2,R = 1.B + ₂B (blue) $₁,R Φ1,Β Φ2,Β - - - - (n₁,R − 1)P₁ + (n2,R − 1)P2 = (N₁,B − 1)P₁ + (n₂,B − 1) P₂ P₁ and P₂ are constant given by radii of curvatures. They do not depend on wavelengths 1 2 - P1 n2,B n2, R P2 n1,B1,R Condition for achromatic doublet: We can write this geometrical ratio in terms of the power at yellow light: Φ1,Υ Vd,1 P1 Q2,Y Va,² ₁,Y/(n₁,x - 1) $2,y/(n2x - 1) P2 Va,1 and Vd,2 are the Abbe numbers for the first lens and the second lens at d wavelength, respectively. since x (n-1) in lens-maker formula
Designing an Achromatic Doublet, 2 • In addition to the required condition shown on previous page, the powers of the two lenses should satisfy our desired overall power : Φ1,γ + Φ2,y = Φ ● Combining the two equations, we can solve for the power for each lens. • The analysis suggest that: O One lens has $₁,y > 0 and is made of a crown glass low dispersion (high Abbe number) - O The other has $2,y < 0 and is made of a flint glass - high dispersion (low Abbe number)
B2. (a) An achromatic doublet has an effective focal length of 200 mm. Find the focal lengths of the individual lenses if they are made of glasses with details shown below. Show your steps in obtaining the values. Neglect the thicknesses of lenses and gaps. Glass 1: Index at d spectral line = 1.639; Abbe number = 55.4 Glass 2: Index at d spectral line = 1.689; Abbe number = 31.2 (6 marks) (b) Sketch your design and indicate how rays parallel to optical axis are focused for each of F, d, and C spectral lines. (3 marks)