Achromatism in lenses

Discussion in 'Physics & Math' started by perfectionist, Dec 10, 2006.

  1. perfectionist Registered Senior Member

    Messages:
    32
    # I have some doubts regarding achromatism of lenses:

    1)If we want to form an achromatic combination of lenses in which both are made of same material, one should be convex and the other should be concave, isn’t it? Is there any other condition to be satisfied along with this?
    2)It is given in a book that, a convex achromatic combination of 2 lenses of the same material placed some suitable distance is possible in the following cases:
    a)Both are convex
    b)Both cannot be concave
    c)Convex lens of greater focal length and concave lens of smaller focal length.
    Is it true?

    3)A convex achromatic combination of 2 lenses of the same material placed in contact can be obtained using a convex lens of lower dispersive power and a concave lens of higher dispersive power. Is it true?

    1)For the first question,I think the reason behind having one convex and other concave in order to form an achromatic doublet is that, they disperse the beam in opposite directions as one is converging and other is diverging. According to the condition for achromatism of lenses,
    w1/w2 = -f1/f2
    as w1 & w2 are different, f1 & f2 should be different. Is it right?

    2)For the second question,
    a) If both are made of same material, then their dispersive powers will be same. From the condition for achromatism of lenses,
    f1=-f2
    If both are convex, then this is not possible, isn't it?
    b) Similarly both cannot be concave.
    c) Similarly this cannot be true.
    So for achromatism of 2 lenses made of same material placed in contact:
    # one should be convex and other should be concave.
    # Both should be of same focal length.

    3) I think the third question is wrong because 2 lenses made of same material cannot have different dispersive powers. Please correct me if I am wrong.
     
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  3. kevinalm Registered Senior Member

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    993
    For #3, but pertains to all three. Two lenses can be of the same material and achromatic if they are not in contact. I think that is what all three questions are leading towards. Iirc, you can balance the dispersion with a convex lense and a stronger negative lense of the same material if they are separated and still have a positive overall focal length.
     
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  5. (Q) Encephaloid Martini Valued Senior Member

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    20,855
    Both lenses can be convex or concave - when both are concave, there use is to increase focal lengths.

    Yes, but they should not be made of the same material or aberrations will occur.

    Not necessarily, even with convex doublets, aberrations are canceled ON-AXIS, however off-axis aberrations may still be present. It would depend on whether or not your application requires good off-axis performance.
     
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  7. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    23,198
    All prior posts seem to agree that one can achieve acromatic image with two lens of the same material (and it may be true, but I do not understand how this can be.)

    If you have two coaxial lenses, L1 & L2 (OF THE SAME MATERIAL, M1) separated by distance D, they will produce a net focal length F1 for red light from point source. (F1 is negative if there is net divergence, by the normal convention.)

    Now keep all geometric things unchanged (surface curvatures, D, etc.) but replace the material so it is now M2, with very different refractive index for red light. Call the the new net focal length F2.

    I claim F1 and F2 are not the same. (Seems so obvious, I will not try to prove it)

    Now supose that the dispersion of M1 is such that for green light the index is the same as for red light in M2.

    Thus when green light is passing thur the first pair of lenses made of M1 the green light will have focus length F2.

    If a given set of lense has F1 for red and F2 for green and if F1 is not the same as F2, then the set of lenses is not acromatic.

    This seems like a rigourous proof that, despite what you book said, it is not possible to make and acromatic net lense for two separate lens OF THE SAME MATERIAL.

    If I am wrong, please point out the error in the above proof.
     
  8. kevinalm Registered Senior Member

    Messages:
    993
    You're thinking of effective focal length, what is called chromatic difference of magnification. And I beleive you are correct. But that is usually considered secondary chromatic abberation. The primary chromatic abberation is the difference between the final focal planes for various colors. And that can be nulled with two widely spaced lenses of the same material, iirc. I've never done the math myself. Simple lens equations won't do, you need to ray trace.
     
    Last edited: Dec 26, 2006
  9. Harmonic_Subset Registered Senior Member

    Messages:
    125
    If I recall correctly, the space between the two lenses is a different material, i.e. air, with a different index of refraction than the two glass lenses. This allows the construction of an achromatic doublet of the same material. Is that right?
     

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