Concave mirrors. Which of the following ray tracings is correct? A: only 1 B: only 2 C: only 3 D: all E: 2& 3

Concave mirrors Which of the following ray tracings is correct? A: only 1 B: only 2 C: only 3 D: all E: 2& 3 1 2 3 c F Point C: geometrical center of the mirror, F: focal point 2

Lens Maker Formula If the front surface of the lens is part of a sphere with radius R 1 and the back surface is part of a sphere with radius R 2, then we can calculate the focal length f of the lens using the lens-makers formula R 1 positive negative positive R 2 negative November 30, 2010 University Physics, Chapter 33 2

The Human Eye (1) The human eye sees by absorbing light Refraction at the cornea and lens surfaces produces a real image on the retina of the eye For an object to be seen clearly, the image must be formed at the location of the retina as shown to the right The shape of the eye cannot be changed so shaping the lens must control the distance of the image cornea The lens is held in place by ligaments that connect it to the ciliary muscle that allows the lens to change shape and thus change the focus of the lens The index of refraction of the two fluids in the eye are close to that of water with a value of 1.44; the index of refraction of the material making up the lens is 1.34 Thus most of the refraction occurs at the air/cornea boundary. retina 33

The Human Eye The extremes over which distinct vision is possible are called the far point and near point The far point of a normal eye is infinity The near point of a normal eye depends on the ability of the eye to focus 5

The Human Eye (3) Several common vision defects result from incorrect focal distances In the case of myopia (near-sightedness), the image is produced in front of the retina In the case of hypermetropia (far-sightedness), the image is produced behind the retina 35

Reading glasses Myopia can be corrected using convex (converging) lenses 7

Systems of Lenses Now we will look at images formed by systems of lenses rather a single lens We use the first lens to image an object We use the image of the first lens as the object for the second lens Thus we can produce various optical instruments with combinations of lenses 30

The Telescope First we will discuss the refracting telescope and then reflecting telescopes The refracting telescope consists of two lenses The objective lens and the eyepiece In our example we represent the telescope using two thin lenses However, an actual refracting telescope will use more sophisticated lenses 42

Geometry of the Telescope Because the object to be viewed is at a large distance, the incoming light rays can be thought of as being parallel (the object is at infinity) The objective lens forms a real image of the distance object at distance f o The eyepiece is placed so that the image formed by the objective is at distance f e from the eyepiece The eyepiece forms a virtual, magnified image of the image formed by the objective The image is at infinity, again producing parallel rays 43

Problems with Refracting Telescopes The objective lens of a refracting telescope is large and heavy The 40-inch refractor at Yerkes weighed 500 pounds Supporting a large glass lens is difficult Must be supported by its edges Constructing large glass lenses is difficult Glass lenses are thick and absorbed light A glass lens has chromatic aberration Different focal lengths for different colors Solution: Replace the objective lens with a mirror 48

The Reflecting Telescope Most large astronomical telescopes are reflecting telescopes with the objective lens being replaced with a concave mirror Large mirrors are easier to fabricate and position than large lenses The eyepiece is still a lens Various types of reflecting telescopes have been developed We will discuss three examples of the geometries of reflecting telescope Reflector Newtonian Cassegrain 49

Basic Reflecting Telescope Basic reflector Replace the objective lens with a parabolic mirror This design is impractical because the observer must be in the line of the incident light 50

Newtonian Reflecting Telescope In 1670 Newton presented his design for a reflecting telescope to the Royal Society The idea for a reflecting telescope came from James Gregory Newton solved the observer problem by placing a small mirror that reflect the light out to an eyepiece This mirror is small compared with the objective mirror and causes only a small loss of light from the image 51

Cassegrain Geometry for Reflecting Telescope A further improvement on the geometry of the reflecting telescope is the Cassegrain geometry (named for the French sculptor Sieur Guillaume Cassegrain) first proposed in 1672 Here a small mirror is used to reflect the image through a hole in the center of the objective mirror This design and many improvements to this basic idea are the basis of modern astronomical telescopes 52

Microscopes exist in many forms The Microscope The simplest microscope is a system of two lenses Objective Lens Eyepiece Object The first lens is a converging lens of short focal length, f o, called the objective lens The second lens is another converging lens of greater focal length, f e, called the eyepiece The object to be magnified is placed just outside the focal length of the objective lens 39

Diffraction and interference 19

Light as Waves We discussed light as rays traveled in a straight line except when they were reflected off a mirror or were refracted at the boundary between two optical media Now we will discuss the implications of the wave nature of light We know that light is an electromagnetic wave (Maxwell s Equations) 1

Ray optics effectively neglects wave properties of light. Addition of waves: subtle issue - PHASE! Recall: laser: 21

Principle of a laser If I add N waves coherently If I add N waves incoherently (random phases) 4 Stot 2 0 2 S = N 2 S 0 4 0 1 2 3 4 5 6 S = NS0 22

Both phase and amplitude are important! I = 4 I0 I = 0 23

Keeping information about the phase of the waves: interference & diffraction Naively, one would expect that using a light of can measure displacements of the order of. λ λ one No! You can do way much better. 24

Tracking phase is a very powerful method Michelson experiment: (difference of speed of light with respect to the ether) few meters, c= 300,000 km/s, travel time ~ 10 nsec V Earth ~ 30 km/s. difference in travel time ~ 1 femto-second late 19th century?! 25

Tracking phase is a very powerful method LIGO - Laser Interferometer Gravitational Wave Observatory -Will measure changes in the distance of mirrors to 10-21 - Displacement of 10-3 of the size of a nucleus over 5 km distance -10-10 of light wavelength 26

Diffraction 27

Ray optics is an approximation According to ray optics (very small wave length) 28

Ray optics is an approximation Diffraction According to ray optics (very small wave length) 28

Diffraction is important for small slits 29

Wave Optics (1) One way to reconcile the wave nature of light with the geometric optical properties of light is to use Huygens Principle developed by Christian Huygens, a Dutch physicist who proposed a wave theory of light in 1678 before Maxwell developed his theories of light This principle states that every point on a propagating wave front serves as a source of spherical secondary wavelets This is (partially) wrong physically, but provides a correct phenomenological description 2

Huygens Constructions (2) We start with a wave front traveling at the speed of light c We assume point sources of spherical wavelets along the wave front These wavelets also travel at c so at a time Δt the wavelets have traveled a distance of cδt If we assume many point sources along the wave front, we can see that the envelope of these wavelets forms a new wave front parallel to the original wave front 5

Wave Optics (2) At a later time, the envelope of these secondary waves becomes a wave front If the original wave has frequency f and speed v, the secondary wavelets have the same f and v 3

Derivation of Snell s Law Use a Huygens construction to derive Snell s Law for refraction between two optical media with different indices of refraction Assume that we have a wave with wave fronts separated by a wavelength λ 1 traveling with speed v 1 in an optically clear medium incident on the boundary with a second optically clear medium as shown 6

Derivation of Snell s Law (2) As the wave moves into the medium, a Huygens wavelet at point e will expand to pass through point c, at a distance of λ 1 from point e. λ 1 h e c The time interval for that is λ 1 /v 1. g In the same time, a wavelet at point h will expand to pass through g, at velocity v 2 and with wave length λ 2. 7

Derivation of Snell s Law (3) The time intervals ec and hg are the same: which we can rewrite as h e g c The wavelengths of the light in the two media are proportional to the speed of light in those media 8

Derivation of Snell s Law (4) We can get a relation between the angle of the incident wave fronts θ 1 with the boundary and the angle of the transmitted wave fronts θ 2 with the boundary by analyzing an expanded region of the Huygens drawing e h c g 9

We can see that Derivation of Snell s Law (5) [hec] Solving for x we get [hcg] h e c Remembering that n = c/v we get g Which is Snell s Law! 10

Diffraction: Sharp Edge Light close to the barrier bends around it and shows interference Described by wavelets close to the barrier Light far from the barrier continues in a straight line 55

Wave properties of light Diffraction through a slit. Antinodal lines can be seen. 39

Different paths d D θ x Dθ 2 θ d/d x d 2 /D λ d λd Condition to see interference patterns 40

Diffraction is important for d λd Fresnel size d D 41

Diffraction is important for d λd Fresnel size d D E.g, for laser λ 10 6 m D 10 m d 3 10 3 m =3mm 41

Diffractive grating d θ d sin θ = mλ 43