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Optics 1 We will cover the basics including reflection, refraction diffuse reflection, index of refraction and handful more of the optics. • Reflection • Diffuse reflection • Refraction • Index of refraction • Speed of light • Snell’s law • Geometry problems

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1. Optics . Reflection .Diffuse reflection . Refraction . Index of refraction . Speed of light . Snell's law Prisms Rainbows Plane mirrors Spherical aberration . Concave and convex mirrors . Focal length & radius of curvature Geometry problems Critical angle .Total internal reflection . Brewster angle Fiber optics .Mirages Dispersion Mirror / lens equation . Convex and concave lenses Human eye Chromatic aberration .Telescopes . Huygens' principle . Diffraction

2. Reflection Most things we see are thanks to reflections, since most objects don't produce their own visible light. Much of the light incident on an object is absorbed but some is reflected. the wavelengths of the reflected light determine the colors we see. When white light hits an apple, for instance, primarily red wavelengths are reflected, while much of the others are absorbed. A ray of light heading towards an object is called an incident ray. If it reflects off the object, it is called a reflected ray. A perpendicular line drawn at any point on a surface is called a normal (just like with normal force). The angle between the incident ray and normal is called the angle of incidence, i, and the angle between the reflected ray and the normal ray is called the angle of reflection, r. The law of reflection states that the angle of incidence is always equal to the angle of reflection.

3. Diffuse Reflection Diffuse reflection is when light bounces off a non-smooth surface. Each ray of light still obeys the law of reflection, but because the surface is not smooth, the normal can point in a different for every ray. If many light rays strike a non-smooth surface, they could be reflected in many different directions. This explains how we can see objects even when it seems the light shining upon it should not reflect in the direction of our eyes. It also helps to Diffuse Reflection (rough sfaces) Specula Reflection

4. Speed of Light & Refraction As you have already learned, light is extremely fast, about 3 x 108 m/s in a vacuum. Light, however, is slowed down by the presence of matter. The extent to which this occurs depends on what the light is traveling through. Light travels at about 3/4 of its vacuum speed (0.75 c) in water and about 2/3 its vacuum speed (0.67 c) in glass. The reason for this slowing is because when light strikes an atom it must interact with its electron cloud, If light travels from one medium to another, and if the speeds in these media differ, then light is subject to refraction (a changing of direction at the interface) Refraction of light Refraction of light waves rays

5. At an interface between two media, both reflection and refraction can occur. The angles of incidence, reflection, and refraction are all measured with respect to the normal. The angles of incidence and reflection are always the same. If light speeds up upon entering a new medium, the angle of refraction, e, will be greater than the angle of incidence, as depicted on the left. If the light slows down in the new medium, g will be less than the angle of incidence, as shown on the right. Reflection & Refraction Incident Ray Incident Ray efracted Ray

6. Index of Refraction, n The index of refraction of a substance is the ratio of the speed in light in a vacuum to the speed of light in that substance: C V Mediunm Vacuum Air (STP) Water (20 C) Ethanol Glass Diamond n- Index of Refraction c Speed of light in vacuum V- Speed of light in medium 1.00029 1.33 1.36 ~1.5 2.42 Note that a large index of refraction corresponds to a relatively slow light speed in that medium

7. Snell's Law Snell's law states that a ray of light bends in such a way that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant. Mathematically, n sine n,sine Here n is the index of refraction in the original medium and n is the index in the medium the light enters. 0, and e, are the angles of incidence and refraction, respectively.

8. Snell's Law Derivation Two parallel ravs are shown. Points A and B are directly opposite one another. The top pair is at one point in time, and the bottom pair after time t. The dashed lines connecting the pairs are perpendicular to the ravs. In time t, point A travels a distance X, while point B travels a distance v. sine-X/d, so x-d sine sine y/d, so yd sine 2 2 Speed ofA: y=x/t Speed of B: V y/t Continued

9. Snell's Law Derivation (cont.) n1 n2 wcsina 1/nsing_ WC sind - .. - - 1 /n sino n, sing = n2 sing

10. Critical Angle The incident angle that causes the refracted ray to skim right along the boundary of a substance is known as the critical angle, ec. The critical angle is the angle of incidence that produces an angle of refraction of 90 . If the angle of incidence exceeds the critical angle, the ray is completely reflected and does not enter the new medium. A critical angle only exists when light is attempting to penetrate a medium of higher optical density than it is currently traveling in. From Snell, n, sine n2 sin 90 Since sin 90 -1, we have n sing- no and the critical angle is

11. Critical Angle Sample Problenm Calculate the critical angle for the diamond-air boundary. Refer to the Index of Refraction chart for the information. ec sin1 (n,/n) = sin-1 (1 / 242) - 24.4 air diamond Any light shone on this boundary beyond this angle will be reflected back into the diamond.

12. Total Internal Reflection Total internal reflection occurs when light attempts to pass from a more optically dense medium to a less optically dense medium at an angle greater than the critical angle. When thi:s occurs there is no refraction, only reflection. n2 >n Total internal reflection can be used for practical applications like fiber optics.

13. Fiber Optics Cont. Fiber optics are often long strands of very pure glass. They are very thin, about the size of a human hair. Hundreds to thousands of them are arranged in bundles (optical cables) that can transmit light great distances. There are three main parts to an optical fiber Core- the thin glass center where light travels. Cladding- optical material (with a lower index of refraction than the core) that surrounds the core that reflects light back into the core. Buffer Coating- plastic coating on the outside of an optical fiber to protect it from damage. Continued...

14. Lingering daylight ater the sun lest ofA Sunlight after Sunset below the horizon is another effect of Apparent refraction. Light travels at a slightly slower speed in Earth's atmosphere than in space. As a result, sunlight is refracted by the atmosphere. In the Actual morning, this refraction causes sunlight position to reach us before the sun is actually above the horizon. In the evening, the position of sun Observer Earth of sun Atmosphere sunlight is bent above the horizon after the sun has actually set. So daylight is extended in the morning and evening because of the refraction of light. Note: the picture greatly exaggerates this effect as well as the thickness of the atmosphere. Different "shapes" of Sun

15. Rainbows A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight incident on a water droplet is refracted. Because of dispersion, each color is refracted at a slightly different angle. At the back surface of the droplet, the light undergoes total internal reflection. On the white 40 violet blue green yellow orango red way out of the droplet, the light is once more refracted and dispersed. Although each droplet produces a complete spectrum, an observer will only see a certain wavelength of light from each droplet. (The wavelength depends on the relative positions of the sun, droplet, and observer.) Because there are millions of droplets in the sky, a complete spectrum is seen. The droplets reflecting red light make an angle of 420 with respect to the direction of the sun's rays; the droplets reflecting violet light make an angle of 40 . Rainbow images

16. Primary Rainbow Primary an RainbowE reen ue Rays from Sun strike falling water droplets 42 for red 40 for violet violet rays from high drops pass overhead violet 51a Ray from Sun violet 42 red red rays from low drops strike below eyes Observer Anti-solar point

17. Secondary Rainbow The secondary rainbow is a rainbow of radius 51 , occasionally visible outside the primary rainbow. It is produced when the light entering a cloud droplet is reflected twice internally and then exits the droplet The color spectrum is reversed in respect to the primary rainbow, with red appearing on its inner edge. ue ow Secondary Rainbow red light from higher drops passes overhead 51 violet light from lower drops bent to point below eye. 420 Observer red

18. Plane Mirror (cont.) Two rays from object P strike the mirror at points B and M. Each ray is reflected such that i r. P' Triangles BPM and BP'M are congruent by ASA (show this), which implies that do o and h- h. Thus, the image is the same distance behind the mirror as the object is in front of it, and the image is the same size as the object. h' Object Image object mage Mirror With plane mirrors, the image is reversed left to right (or the front and back of an image ). When you raise your left hand in front of a mirror, your image raises its right hand. Why aren't top and bottom reversed?

19. Concave Mirrors Concave mirrors are approximately spherical and have a principal axis that goes through the center, C, of the imagined sphere and ends at the point at the center of the mirror, A. The principal axis is perpendicular to the surface of the mirror at A CA is the radius of the sphere,or the radius of curvature of the mirror, R .Halfway between C and A is the focal point of the mirror, F. This is the point where rays parallel to the principal axis prindpal will converge when reflected off the mirror. R. Aris The length of FA is the focal length, f The focal length is half of the radius of the sphere (proven on next slide)

20. Spherical Aberration Spherical Mirror Parabolic Mirror Only parallel rays close to the principal axis of a spherical mirror will converge at the focal point. Rays farther away will converge at a point closer to the mirror. The image formed by a large spherical mirror will be a disk, not a point. This is known as spherical aberration. Parabolic mirrors don't have spherical aberration. They are used to focus rays from stars in a telescope. They can also be used in flashlights and headlights since a light source placed at their focal point will reflect light in parallel beams. However, perfectly parabolic mirrors are hard to make and slight errors could lead to spherical aberration Continued.

21. Spherical vs. Parabolic Mirrors Parallel rays converge at the focal point of a spherical mirror only if they are close aberration. The mirror focuses all parallel to the principal axis. The image formed in a large spherical mirror is a disk, nota point (spherical aberration). Parabolic mirrors have no spherical rays at the focal point. That is why they are used in telescopes and light beams like flashlights and car headlights. R S

22. Concave Mirrors: Object beyond C The image formed when an object is placed beyond C is located between C and F. It is a real, inverted image that is smaller in size than the object. object mage Animation 1 Animation 2

23. Concave Mirrors: Object at C or F What happens when an object is placed at C? The image will be formed at C also, but it will be inverted It will be real and the same size as the object Object Image What happens when an object is placed at F? No image will be formed. All rays will reflect parallel to the principal axis and will never converge. The image is "at infinity,"" Object No hmage is Fonmed

24. Convex Mirror Diagram The image formed by a convex mirror no matter where the obiect is placed will be virtual, upright, and smaller than the object. As the object is moved closer to the mirror, the image will approach the size of the object. object image

25. Mirror Sign Convention f= focal length d image distance do object distance + for real image for virtual image + for concave mirrors for convex mirrors

26. Magnification By definition, m magnification h,- image height (negative means inverted) ho- object height Magnification is simply the ratio of image height to object height. A positive magnification means an upright image.

27. Magnification Identity: To derive this let's look at two rays. One hits the mirror on the axis. The incident and reflected rays each make angle 0 relative to the axis. A second ray is drawn through the center and is reflected back on top of itself (since a radius is always perpendicular to an tangent line of a circle). The intersection of the reflected rays determines the location of the tip of the image. Our result follows triangles, with the negative sign a consequence of our sign convention. (In this picture h, is negative and d is positive.) object from similar image, height

28. Convex Lens: Object Beyond 2F he image formed when an object is placed beyond 2F is located behind the lens between F and 2F. It is a real, inverted image which is smaller than the object itself. object 2F 2F image

29. Concave Lenses Rays traveling parallel to the principal axis of a concave lens will refract as if coming from the Pfocus. Rays traveling toward the focus will refract parallel to the principal axis (-L . Rays traveling directly through the center of a concave lens will leave the lens traveling in the exact same direction, just as with a convex lens. F 2F

30. Lens Sign Conventio f= focal length d image distance do object distance + for real image for virtual image + for convex lenses for concave lenses

31. Convex Lens Made of Water Glass Since water has a higher index of refraction than air, a convex lens made of water will converge light just as a glass lens of the same shape. However, the glass lens will have a smaller focal length than the water lens (provided the lenses are of same shape) because glass has an index of refraction greater than that of water. Since there is a bigger difference in refractive index at the air-glass interface than at the air-water interface, the glass lens will bend light more than the water lens. Air n 1.5 H,O Air n 1.33

32. Chromatic Aberration As in a raindrop or a prism, different wave-lengths of light are refracted at different angles (higher frequency Blue Red greater bending). The light passing through a lens is slightly dispersed, so objects viewed through lenses will be ringed with color. This is known as chromatic aberration and it will always be present when a single lens is used. Chromatic aberration can be greatly reduced when a convex lens is combined with a concave lens with a different index of refraction. The dispersion caused by the convex lens will be almost canceled by the dispersion caused by the concave lens. Lenses such as this are called achromatic lenses and are used in all precision optical instruments. Chromatic Aberration Red Blue Achromatic Lens

33. The first eye shown suffers from farsightedness, which is also known as hyperopia. This is due to a focal length that is too long, causing the image to be focused behind the retina, making it difficult for the person to see close up things. Hyperopia The second eye is being helped with a convex lens. The convex lens helps the eye refract the light and decrease the image distance so it is once again focused on the retina Formation of image behind the retina in a hyperopic eye. Hyperopia usually occurs among adults due to weakened ciliary muscles or decreased lens flexibility, Convex lens correction for hyperopic eye Farsighted means "can see far" and the rays focus too far from the lens.

34. The first eye suffers from nearsightedness, or myopia. This is a result of a focal length that is too short, causing the images of distant objects to be focused in front of the retina. Myopia The second eye's vision is being corrected with a concave lens. The concave lens diverges the light rays, increasing the image distance so that it is focused on the retina. Formation of image in front of the retina in a myopic eye. Nearsightedness is common among young people, sometimes the result of a bulging cornea (which will refract light more than normal) or an elongated eyebal Concave lens correction for myopic eye. Nearsighted means "can see near" and the rays focus too near the lens.

35. Refracting Telescopes Refracting telescopes are comprised of two convex lenses. The objective lens collects light from a distant source, converging it to a focus and forming a real, inverted image inside the telescope. The objective lens needs to be fairly large in order to have enough light-gathering power so that the final image is bright enough to see. An eyepiece lens is situated beyond this focal point by a distance equal to its own focal length. Thus, each lens has a focal point at IF. The rays exiting the eyepiece are nearly parallel, resulting in a magnified, inverted, virtual image. Besides magnification, a good telescope also needs resolving power, which is its ability to distinguish objects with very small angular separations. Convex Lens Refracting Telescope Eye Eyepiece Lens Objective Lens Focal Length

36. Reflecting Telescopes Galileo was the first to use a refracting telescope for astronomy. It is difficult to make large refracting telescopes, though, because the objective lens becomes so heavy that it is distorted by its own weight. In 1668 Newton invented a reflecting telescope. Instead of an objective lens, it uses a concave objective mirror, which focuses incoming parallel rays. A small plane mirror is placed at this focal point to shoot the light up to an eyepiece lens (perpendicular to incoming rays) on the side of the telescope. The mirror serves to gather as much light as possible, while the eyepiece lens, as in the refracting scope, is responsible for the magnification. Reflection Reflecting Telescope Focus Focal Length

37. Double Slit (cont.) screen In practice, L is far greater than a, meaning that segments measuring d, and d2 are virtually parallel. Thus, both ray:s make an angle relative to the vertical, and the bottom right angle of the triangle is also (just like in the single slit case). This means the extra distance traveled is given by a sine. Therefore, the required condition for a bright spot at P is that there exists a natural number, n, such that: 2 a sin nn If white light is shone at the slits, different colors will be in phase at different angles Electron diffraction