Question

Two thin lenses of same material and mounted coaxially are to have minimum spherical and chromatic aberrations. Find ratio of their local length.

Answer 1

1

Answer 2

To minimize both spherical and chromatic aberrations for a system of two thin lenses made of the same material and mounted coaxially, we consider the conditions for each type of aberration.

1. Condition for Minimum Chromatic Aberration: For two thin lenses of focal lengths $f_1$ and $f_2$, made of materials with dispersive powers $\omega_1$ and $\omega_2$ respectively, and separated by a distance $d$, the condition for minimum (zero) chromatic aberration (achromatism) is given by:

   $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} - \frac{d(\omega_1 + \omega_2)}{f_1 f_2} = 0$

   The question states that the lenses are made of the same material. Therefore, their dispersive powers are equal: $\omega_1 = \omega_2 = \omega$. Substituting $\omega_1 = \omega_2 = \omega$ into the condition:

   $\frac{\omega}{f_1} + \frac{\omega}{f_2} - \frac{d(\omega + \omega)}{f_1 f_2} = 0$

   $\omega \left( \frac{1}{f_1} + \frac{1}{f_2} - \frac{2d}{f_1 f_2} \right) = 0$

   Since $\omega \neq 0$, we can divide by $\omega$:

   $\frac{1}{f_1} + \frac{1}{f_2} - \frac{2d}{f_1 f_2} = 0$

   Multiplying by $f_1 f_2$:

   $f_2 + f_1 - 2d = 0$

   $d = \frac{f_1 + f_2}{2}$

   This is the condition for minimum chromatic aberration when two lenses made of the same material are separated by a distance $d$.

2. Condition for Minimum Spherical Aberration: For a system of two lenses, spherical aberration is minimized when the deviation of light rays is distributed symmetrically between the lenses. This is generally achieved when the lenses are identical in shape and power. Therefore, for minimum spherical aberration, the focal lengths of the two lenses should be equal:

   $f_1 = f_2$

3. Combining Both Conditions: To achieve minimum spherical and chromatic aberrations simultaneously, both conditions must be satisfied. From the spherical aberration condition, we have $f_1 = f_2$. Substitute $f_2 = f_1$ into the chromatic aberration condition $d = \frac{f_1 + f_2}{2}$:

   $d = \frac{f_1 + f_1}{2} = \frac{2f_1}{2} = f_1$

   So, for minimum spherical and chromatic aberrations, the two lenses must have equal focal lengths ($f_1 = f_2$) and be separated by a distance equal to their common focal length ($d = f_1$).

   The question asks for the ratio of their focal lengths, which is $\frac{f_1}{f_2}$. Since $f_1 = f_2$, the ratio is:

   $\frac{f_1}{f_2} = \frac{f_1}{f_1} = 1$

For two lenses of the same material, minimum chromatic aberration occurs when the separation $d = (f_1+f_2)/2$. Minimum spherical aberration for a two-lens system is achieved when the lenses are identical, i.e., $f_1=f_2$. Combining these conditions, we get $d=f_1$. Thus, for minimum spherical and chromatic aberrations, the focal lengths must be equal, leading to a ratio of 1.