Focal Length of a Lens as a function of light frequency

How does the focal length of a glass lens for blue light compare with that for red light? Consider the case of either a diverging lens or a converging lens.

 

This question really has three parts:

  • Focal Length of a lens
  • Effect of light frequency (color)
  • Diverging and Converging lens

 

Focal Length of the Converging and Diverging Lens

For the converging and diverging lens, the focal point has a different meaning. First, consider the converging lens. Parallel rays entering a converging lens will be brought to focus at the focal point F of the lens. The distance between the lens and the focal point F is called the focal length, f. The focal length is a function of the radius of curvature of both sides or planes of the lens as well as the refractive index of the lens. The formula for focal length is below,
(1/f) = (n-1)((1/r1)-(1/r2)).

This formula also works for a diverging lens, however the directions of the radius of curvature must be taken into account. If for instance the center of the circle for one side of the lens is to the left of the lens, one may chose that direction to be positive and the other direction to be negative; as long as one maintains the same standard for direction.

converg

If the focal length of a lens is negative, meaning that the focal point is behind the lens, on the side at which the rays entered, this is a diverging lens.

diverg

 

Interaction of Color with Focal Length

The other part of this question dealt with how the focal length would change for one color such as blue versus another color such as red. The key to this relationship is the refractive index of the lens, as the refractive index can change with regards to the color (i.e. frequency).

The material from which the lens is made is not known, however as demonstrated by the following table, the refractive index is consistently higher for smaller wavelength colors.

33

Reviewing the focal length formula, it is understood from the inverse proportionality of the equation that as the refractive index increases, the focal length will decrease. Blue has a higher refractive index than red. Therefore, blue will have a smaller focal length than red.

focallength2

 

 

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