VII–2 Basic Optical Elements and Instruments

1. VII–2 Basic Optical Elements and Instruments

2. Main Topics • Refraction, Dispersion and Refraction Optics. • Thin Lenses. Types and Properties. • Combination of Lenses. • Basic Optical Instruments • Human Eye • Magnifying Glass • Telescope • Microscope

3. Refraction I • Another important basic optical effect is refraction. This time rays pass from one material to another. Transparent materials differ in, so called optical density, the more dense material the lower is the speed of light in it. We characterize this by the absolute refraction index: n = c/v where c is the speed of light in vacuum and v speed in the particular material.

4. Refraction II • We can again use the Fermat’s principle to find the law of refraction. • To find which ray makes it first from S to P is a similar problem as if we want to safe a drowning person and we optimize for the shortesttime, taking into account that we run much faster than swim.

5. Refraction III • We use the general definition that the correct ray is a stationary one. In other words, if we take some close ray its time of flight will be roughly the same. • Let the point S be in a space where the light travels with the speed v1 = c/n1 and P in the space where the speed is v2 = c/n2.

6. Refraction IV • Now, let the SCP be the correct ray for and the SXP some neighbor ray. Should the time of flight be the same: EC/v1 = XF/v2 • We use : EC = XCsin1 and XF = XCsin2 substitute for v1 and v2 and get the • Snell’s law: n1sin1 = n2sin2

7. Refraction V • We see that the higher is the optical density or the slower is the speed of light the smaller is the refraction angle. • If the angle of incidence from the less dense material is 90° the refracted angle is given: • sin2 = n1/n2 the maximum reflected angle or the critical angle.

8. Refraction VI • If the beam would try to pass from the optically dense material under an incident angle higher than this critical angle it would not get through the boundary but rather be totally reflected. • The effect of total (internal) reflection is used in fiber optics.

9. Dispersion I • Transparent materials have an interesting property that the speed of light and thereby their refractionindexdepend on the wavelength of the applied light. • This means that light of every wavelength of color is refracted under a (little) different angle.

10. Dispersion II • The effect of dispersion complicates design of optical systems. • On the other hand it gives us the possibility to decompose the visible light and near IR and UV regions into different wavelengths which has a great impact for instance on studies of properties of matter by spectroscopicmethods.

11. Refraction Optics I • The effect of refraction is used to build optical components and systems. • If we have a point S in the medium n1 and the point P in the medium n2 > n1 we may use the Fermat’s principle to find the shape of the boundary between the media so the points are conjugated or the optical system is stigmatic for them.

12. Refraction Optics II • If we compare some refracted ray with the one directly connecting both points we find a relation: l1n1 + l2n2 = s1n1 + s2n2 • The corresponding surface is of the fourth order, so called, Cartesian ovoid. • We readily understand from here, why the denser media must be convex.

13. Refraction Optics III • If we move one of the points S or P into infinity the surface becomes second order, either elliptical or hyperbolical. • This can be in principal used to construct lenses - optical components from some material, which allow that the object as well as the image are in the same media.

14. Refraction Optics IV • Ideal lenses are for instance double hyperbolic or planar-hyperbolic. • Although, recently they can be, in principle, machined, for the same reasons, which were described in the case of mirrors aspherical surfaces are approximated by spherical ones. Again they can be successfully used only in the paraxial region.

15. Thin Lenses I • Very important lenses are those which can be considered as thin. • They can be characterized by a single parameter the focal length f. It is the distance from the center of the lens to the focal point F. It is the point in which the rays approaching the lens in parallel with the optical or principalaxis meet behind it.

16. Thin Lenses II • To understand many optical instruments it is good to remember that also other parallel rays which fall at the lens at an angle focus in one point, which lays in the focal plane of the lens. • Optometrist and ophthalmologist use the powerP = 1/f to specify lenses. Its unit is diopter (D), 1D = 1m-1.

17. Thin Lenses III • The, so called, lensmaker’s equation can be derived which relates the focal distance of a thin length with the radii of its surfaces • 1/f = (n-1)(1/R1 + 1/R2) • Appropriate signconventions must be obeyed. • Note that the focal length is the same from both sides even if the radii are different.

18. Thin Lenses IV • As it was the case of the mirrors, lenses can be converging and diverging and images can be real and virtual. • To find an image of some object, we can again use two of three special rays. We can employ the properties of a focal point and the fact that the beam passing through the optical center is not deflected.

19. Thin Lenses V • The lens equation which relates the distances of the object and image with the focal distance can be easily derived: 1/do + 1/di = 1/f • and lateral magnification is defined as the ratio of the image height to the object height m = ho/hi = - di/do

20. Combination of Lenses • We start from the lens closest to the object. • We display the object as if only this lens were present. • The image of produced by the first lens will be the objectfor the second lens. • Then we display the new object by the second lens only. And so on.

21. The Human Eye I • Most of the focusing (refraction) is done by the cornea (n = 1.376). The lens does just the ‘fine tuning’. • The quality of focusing and the depth of focus depends on the iris. The smaller the aperture the better. • Normal eye had the near point at 25 cm and the far point in infinity.

22. The Human Eye II • In the case of nearsightedness (myopia) the far point is not infinity. This has to be corrected by a diverging lens. • In the case of farsightedness (hyperopia or presbyopia – developed by age) the eye can’t focus on near objects. This has to be corrected by a converging lens.

23. The Human Eye III • The eye is relaxed if it watches the far point so eyepieces usually produce parallel rays. • Some other optical instruments produce a virtualimage in the conventionallength equal to the standard near point at 25 cm.

24. Magnifying Glass • Magnifying glass is used: • either the object is in the focal plane and we watch by relaxed eye. • or the lens is close to the eye (Sherlock Holmes) and a virtual image is produced in the conventional distance. • Magnification is the angle magnification – we see objects as big as is the angle on retina.

25. Telescopes • Astronomical telescope are two lenses an objective (longer f) and an eyepiece which share the same focal plane. The angle magnification is ratio of the focal lengths. • Important are reflecting telescopes: • Large mirrors are easier to produce and support • Mirrors don’t suffer from color aberration.

26. Compound Microscope • The principle of a microscope can be shown also using twolenses. The objective (now with very short f) produces a real image. It is watched by the eyepiece, which usually produces the imaginary image in the conventional distance. • Good microscopes are complicated since it is important to compensateaberrations.

27. Homework • The last homework is due tomorrow!

28. Things to read and learn • Chapters 33, 34 • Please, read and try to understand even the parts which were not dealt with in detail in the lecture. You should have far enough background knowledge to understand everything!

29. Lens Equation • . ^

30. Maxwell’s Equations I • . ^