Design Achromatic Lens: Crown & Flint Glass, Focal Length 100cm

[Guji_Gyal]

Hey guys and gals...i am really stuff on dis question.. n i really need 2 knw homework 2 do dis.. so ne1 please help..n post der views...cheers!

Design a converging achromatic lens of focal length 100cm. Refractive indices and dispersive powers for the two glasses available are 1.51 and 0.016 for crown glass and 1.61 and 0.026 for flint glass. The converging element of the achromat is to be bi-convex and the diverging element is to be plano-concave. Which glass is to be used for each of the two elements of the lens, and determine the radii of curvature of the lens surfaces?

 

[Andrew Mason]

Hey guys and gals...i am really stuff on dis question.. n i really need 2 knw homework 2 do dis.. so ne1 please help..n post der views...cheers!

Design a converging achromatic lens of focal length 100cm. Refractive indices and dispersive powers for the two glasses available are 1.51 and 0.016 for crown glass and 1.61 and 0.026 for flint glass. The converging element of the achromat is to be bi-convex and the diverging element is to be plano-concave. Which glass is to be used for each of the two elements of the lens, and determine the radii of curvature of the lens surfaces?

The diverging element would be the higher density glass. The dispersion in the concave lens has to correct for the dispersion in the double convex lens. With the higher index, it can do this with a lower power lens. Since the powers add and the power of the concave is negative, the result is a + power (converging) lens.

The rest of it is a little complicated.

The power / focal length of the biconvex lens, with surfaces having radii R_a and R_b, would be determined by the lensmaker's formula:

$$P_1 = \frac{1}{f_1} = (n_1-1)(\frac{1}{R_a}+ \frac{1}{R_b}) = (n_1 -1)k_{ab}$$

where $k_{ab} = (\frac{1}{R_a}+ \frac{1}{R_b})$

and the power/focal length of the concave surface (radius = -R_b) would be:

$$P_2 = \frac{1}{f_2} = (n_2-1)\frac{1}{-R_b} = (n_2-1)k_c$$

where $-R_b = 1/k_c$ is the radius of curvature of the concave surface

The focal length of the combined lens is

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

(where f_2 is negative).

The key to this problem is relating the dispersion to the radius of the lens.
For the convex lens the dispersion is:

$$d_{ab} = d_1k_{ab}$$ where $d_1$ is the dispersive power of the crown glass

For the concave lens the dispersion is:

$$d_c = d_2k_c$$ where $k_c$ is negative. $d_2$ is the dispersive power of the flint glass

The condition for 0 dispersion is $d_{ab} + d_c = 0$

So try to work out the values of $R_a and R_b$ from all that!

AM

 

[thepatientmental]

Hi there,

Designing an achromatic lens can seem intimidating, but with some basic principles and equations, we can easily solve this problem. Let's break it down step by step.

First, we need to understand what an achromatic lens is. An achromatic lens is a type of lens that is designed to minimize chromatic aberration, which is the distortion of colors in an image. This is achieved by using two different types of glass with different dispersive powers, which means they bend light at different rates. These two glasses are called crown glass and flint glass.

Now, let's move on to the design of the lens. Since we want a converging lens, the first element of the lens, the one that will bend light towards a focal point, should be made of flint glass. This is because flint glass has a higher refractive index and dispersive power, which will help in achieving the desired focal length of 100cm.

Next, we need to determine the radii of curvature for both elements of the lens. To do this, we can use the lensmaker's equation: 1/f = (n-1)(1/R1 - 1/R2), where f is the focal length, n is the refractive index of the material, and R1 and R2 are the radii of curvature for the two lens surfaces.

For the converging element, we know that the focal length is 100cm and the refractive index of flint glass is 1.61. So, we can rearrange the equation to solve for R1 and R2. Plugging in the values, we get R1 = 50cm and R2 = 50.5cm.

For the diverging element, we know that the focal length is also 100cm, but the refractive index of crown glass is 1.51. Using the same equation, we get R1 = -50cm and R2 = infinity (since a plano-concave lens has one flat surface).

So, to summarize, we need a bi-convex lens with a radius of curvature of 50cm for the converging element, made of flint glass, and a plano-concave lens with a radius of curvature of -50cm, made of crown glass.

I hope this helps with your homework and gives you a better understanding of how to design an achromatic lens. Good luck!