Optical Instruments

Formulas:

RP = 1/f Relation of Refractive Power to focal length M = q'/q M = 1 + N/f Magnifying glass with eye focused at near point M = N/f Magnifying glass with eye focused at infinity sinq = 1.22l/D q = 1.22l/D For small angles.

1. Optical Instruments

   1.1 The Human Eye

       In the human eye, most refraction takes place at the air, cornea interface, not in the lens. The Far point of the eye is the farthest point an object can be from the eye, and that a fully relaxed eye can focus on. The Near point of the eye is the nearest point that a fully tensed eye can focus on. (about 25 cm, i.e. 10 inches).

       The lens can tense up and grow thicker which changes the incident angle and thus changes the amount of diffraction.

       A Nearsighted eye can focus on near objects, but not far. Lenses must change the image distance from infinity to the far point of the eye. A Farsighted eye can see far but not near. Lenses must change the image distance from too close to the near point of the eye.

Problem 1: A nearsighted person wears glasses which have a focal length of -20.0 cm. The lens is 1.8 cm from the eye. If the person changes to contact lenses what should the focal length be? 1/¥ + 1/di = -1/20 di= -20 cm. So the far point of the eye is 20.0 + 1.8 = 21.8 cm. For contacts, then 1/¥ - 1/21.8 = 1/f,  so f = -21.8. Opthamologists, Optometrists, and Opticians write prescriptions in diopters. The refractive power of a lens in diopters = 1/f (in meters)

   1.2 The Magnifying Glass

       For optical instruments we measure their magnifying power using something called the angular magnification. This is defined as the angular size of final object divided by angular size of initial object when the object is initially at the near point of the eye. See figure 25-13 in the book.

M = q'/q

       For different optical instruments the formula for how the angular magnification relates to the position of the lenses will change.

   For a magnifying glass we find
 

M = 1 + N/f eye focused at near point  M =N/f eye focused at infinity

   1.3 Lens Aberrations

       Lenses do not produce perfect images. The imperfections are called aberrations and come from a number of sources

   1.3.1 Spherical Aberrations.

       Light at the edges of a lens does not focus at the focal point of the lens. Thus, the image is not perfectly focused. We can minimize this by blocking the light at the edges of the lens.

   1.3.2 Chromatic Aberrations

       Because light of different wavelengths have different indices of refraction, the light of different wavelengths focus at different places. This is chromatic aberration, and can be reduced by using a compound lens as shown in the figure.

   1.3.3 Resolving Power

       Suppose I have two closely spaced objects. If they are far apart, they look like two objects, but when they are close together, we cannot resolve them. They look like one object. One criteria to determine whether or not they can be resolved is called Rayleigh's Criteria. It states two point objects are just resolved when the first dark fringe in the diffraction pattern of one falls directly on the central bright fringe in the diffraction pattern of the other.

       This occurs when sinq= 1.22l/D where D is the diameter of the circular opening trying to resolve the objects. For small angles sinq »q when q is expressed in radians. So, Rayleigh's criteria becomes

q = 1.22 l/D (q in radians)

PROBLEM 2: Two motorcycles, separated by 2.00 m are approaching an observer holding an infrared detector which is sensitive to radiation with a wavelength of 885 nm. What aperture diameter is required if the detector is to resolve the headlights at a distance of 10.0 km? q = 1.22 l/D = d/L D = 1.22lL/d = 1.22(885×10-9m)(10×10-3 m)/2.00 m = 5.40×10-3 m

   1.3.4 X-ray Diffraction

       What limits how closely I can resolve details of an object? Certainly lens aberration and diffraction as stated above. But also, you need to probe details of an object with radiation that has a wavelength less than the details you are trying to resolve. Consequently, visible light with a wavelength of a few hundred nanometers cannot resolve details of any object that are smaller than a few hundred nanometers. To resolve smaller details, we need a probe with a shorter wavelength. To resolve crystal structures in matter we use X-rays.

       Diffraction patterns can be created by any regular patterns. In particular, the crystal structure of many solids has a lattice spacing about the same size as the wavelength of x rays. So we can see the diffraction pattern from a crystal structure when we shoot x-rays through them.