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Separation of two achromatic lenses

Short answer

For same material, achromatism and minimum spherical aberration require x = (f1+f2)/2 = f1 - f2, giving f1 = 3f2. Equivalent focal length F = 1.5 f2, so with F = 50 cm, f2 = 33.3 cm, f1 = 100 cm, separation x = 66.7 cm.

Two thin lenses of focal length f1 and f2 separated by a distance having an equivalent focal length 50 cm. The combination satisfies the condition for no chromatic aberration and minimum spherical aberration. Find the separation between the two lenses if both lenses are of same materials.

Conditions for a two-lens combination of the same material

  1. No chromatic aberration (achromatism): ω/f₁ + ω/f₂ = 2xω/(f₁f₂), which gives x = (f₁ + f₂) / 2.
  2. Minimum spherical aberration: x = f₁ − f₂ (with f₁ > f₂).

Step 1: relate f₁ and f₂

(f₁ + f₂)/2 = f₁ − f₂
f₁ + f₂ = 2f₁ − 2f₂
f₁ = 3 f₂

Step 2: equivalent focal length

1/F = 1/f₁ + 1/f₂ − x/(f₁f₂)
F   = f₁f₂ / (f₁ + f₂ − x)
    = f₁f₂ / ((f₁ + f₂)/2)       (since x = (f₁+f₂)/2)
    = 2 f₁f₂ / (f₁ + f₂)

With f₁ = 3f₂: F = 2 × 3f₂² / 4f₂ = 1.5 f₂.

Step 3: numbers (F = 50 cm)

f₂ = 50/1.5 = 33.33 cm
f₁ = 3 f₂   = 100 cm
x  = (f₁ + f₂)/2 = (100 + 33.33)/2 = 66.67 cm

Check: x = f₁ − f₂ = 66.67 cm, and F = 100 × 33.33 / (133.33 − 66.67) = 50 cm.

Answer: the two lenses must be 66.7 cm (about 2/3 m) apart (f₁ = 100 cm, f₂ = 33.3 cm).

✓ Answer checked by Claude Opus 5.5 (high effort)

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