Geometric Optics

1. Geometric Optics Unit 10

2. Optics • We have already seen that light can be thought of as an EM wave. • We perceive light using our eyes, and depend heavily on sight. • For this reason, the study of light and how it behaves is an important one. • This field is called optics.

3. Optics • We can perceive an object by sight in two ways: • The object may be a source of light (like a light bulb, fire, or a star). • We see the light that is reflected off of the object. • The subject of how objects emit light is rather complicated. • We will look at how light is reflected (and transmitted).

4. Geometric Optics • Although light is an EM wave, many of its most important properties can be understood without worrying about this fact. • These include reflection, refraction, and the formation of images using mirrors and lenses. • This area of optics is called geometric optics.

5. The Ray Model of Light • When the wave characteristics of light can be ignored, we employ the ray model to describe the propagation of light. • This model assumes that light travels in straight lines. • These lines are represented with rays.

6. The Ray Model of Light • The ray itself is actually an idealization. • It represents an extremely narrow beam of light. • The ray points along the direction the EM wave is moving. • Therefore, it is perpendicular to both the E and B fields of the wave.

7. The Ray Model of Light • In the ray model, rays are reflected off of each point on the object is many different directions. • Some of these rays reach our eyes.

8. The Ray Model of Light • When we see an object, light is reaching out eyes from every point on the object. • This also explains why we can’t see the far side of the object: the light rays can’t reach our eye since it only travels in straight lines.

9. Thoughts on Geometric Optics • All phenomena covered in geometric optics can be described using light rays. • In our pictures, we will generally only draw the rays that reach our eyes. • Because these explanations involve straight lines at various angles, these are all essentially geometry problems.

10. Reflection

11. Reflection • When light strikes the surface of an object, at least some of that light is reflected off of the object. • The rest of the light is either transmitted through the object, or absorbed by the object (and transformed into heat). • For very shiny objects (such as mirrors) over 95% of the light is reflected.

12. Reflection • Let’s consider a narrow beam of light striking a flat surface. • The beam is represented by a light ray, called the incident ray.

13. Reflection • The ray makes an angle with the line perpendicular to the surface. • This angle is called the angle of incidence and is represented by θi.

14. Reflection • The incident ray is reflected off the surface and travels off in a new direction. • This ray is called the reflected ray.

15. Reflection • The reflected ray also makes an angle with the line normal to the surface. • This angle is called the angle of reflectionand is represented by θr.

16. Reflection • How are these two angles related? • The answer is given by the law of reflection. • This is the same law we saw for 2D and 3D waves last quarter.

17. Law of Reflection • The law of reflection states The angle of reflection equals the angle of incidence. θr= θi

18. Law of Reflection • Furthermore, both the incident and reflected rays lie in the same plane. • It is also the same plane as the line normal to the surface.

19. Surface Types • What happens if the surface is not flat, but is rough like the surface of water (or even a piece of paper)? • The law of reflection still applies at each point along the surface. • However, since the surface is rough, it is tilted different ways at different points.

20. Diffuse Reflections • As a result, when a beam of light hits a rough surface, the light is reflected in many different directions depending on where it hits the surface. • This is called diffuse reflection.

21. Diffuse Reflections • When you look at light reflected off of a rough surface, it is possible to see the image from many different angles. • This is because your eye is picking up different reflected rays at different points.

22. Diffuse Reflections • When you read a book or look at other everyday objects, you are seeing diffuse reflections. • You can see these objects from many different angles because light is reflected at many different angles off of the rough surfaces.

23. Specular Reflections • Reflections off of flat, reflective surfaces are called specular reflections. • This comes from the Latin word for mirror, “speculum.”

24. Specular Reflections • When you shine a beam of light on a mirror, you will only be able to see the reflection if your eye is at the correct location. • This position is determined by the law of reflection.

25. Plane Mirrors

26. Plane Mirrors • We are most familiar with plane mirrors. • A plane mirror is a surface that is essentially flat and highly reflective. • Generally, a plane mirror is made by putting a metallic coating on one side of a flat piece of glass.

27. Plane Mirrors • When you look in a mirror, you see yourself and the objects around you. • It appears as if what you are seeing is in front of you, beyond the plane of the mirror. • This is not really the case. • What you are seeing is called an image of the objects that are in front of the mirror.

28. Example: Plane Mirror • Let’s consider the case of viewing a simple object in a mirror. • When light strikes the bottle, light rays are reflected off in all directions. • However, only some of these rays enter your eye.

29. Example: Plane Mirror • This bundle of rays are bordered by the four rays shown in the diagram. • These light rays reflect off the mirror and enter the eye.

30. Example: Plane Mirror • However, the eye interprets these rays to have traveled in a straight line from a single point in space. • This point is called the image point.

31. Example: Plane Mirror • Let’s focus on the two rays that originate from point A. • These rays hit the mirror at B and B’. • From the geometry of the problem, we can see that angles ADB and CDB are right angles.

32. Example: Plane Mirror • Lastly, we know that, by the law of reflection, θi = θr. • Based on this, angles ABD and CBD must be equal. • Thereforce, AD = CD.

33. Example: Plane Mirror • In other words, the image appears as far as behind the mirror as the object is in front. • The distance from the object to the mirror is called the object distance, do.

34. Example: Plane Mirror • The distance from the image to the mirror is called the image distance, di. • Then, for a plane mirror

35. Example: Plane Mirror • Notice also that the height of the image is the same as the height of the object. • Also notice that no actual light rays pass through the image location (it’s behind the mirror). • Images of this type are called virtual images.

36. Example: Plane Mirror • In other situations (such as with lenses), real light rays will pass through the image location. • These images are called real images. • Our eyes can see both types of images. It is up to use the geometry of the system to identify the image type.

37. Homework • Read 23-1 and 23-2. • Do problems 1, 2, and 4 on page 658.

38. Spherical Mirrors

39. Spherical Mirrors • More often than not, reflecting surfaces are not flat. • Often, spherical mirrors are used to magnify objects, or to obtain a wider field of view.

40. Spherical Mirrors • There are two types of spherical mirrors. • A convex mirror has the reflective surface on the the outside of the sphere. • A concave mirror has the reflective surface on the inside of the mirror.

41. Spherical Mirrors • There are two types of spherical mirrors. • A convex mirror has the reflective surface on the the outside of the sphere. • A concave mirror has the reflective surface on the inside of the mirror.

42. Spherical Mirrors • In both cases, the law of reflection is obeyed at each point on the surface. • Since the surface is curving, not all incoming rays are reflected at the same angle. • However, if the curvature is known, we can predict how the rays will be reflected.

43. Properties of Spherical Mirrors • To figure out how images are formed with spherical mirrors, we need to define some characteristics of the mirror. • Let’s consider rays from a distant object striking a concave mirror.

44. Properties of Spherical Mirrors • If the object is very far away, the rays arriving at the mirror are nearly parallel to one another. • For our purposes, we will consider these rays to be exactly parallel (far away objects like the sun approach this).

45. Properties of Spherical Mirrors • As the parallel rays arrive at the mirror, they are reflected according to the law of reflection at each point on the surface. • Since the mirror is curved, the rays do not come together at a single point (recall that the rays must come together to form a sharp image).

46. Properties of Spherical Mirrors • However, if the size of the mirror is small compared to its curvature, then all the rays will come together at almost the same point. • This point is known as the focus.

47. Properties of Spherical Mirrors • The line going through the center of the mirror is known as the principal axis of the mirror. • The mirror is symmetric about this axis. • All rays entering the mirror parallel to the principal axis are reflected through the focus.

48. Properties of Spherical Mirrors • For this reason, this point F is also called the focal point of the mirror. • The distance between F and the center of the mirror (A) is called the focal length, f, of the mirror. • The focal point is also the image point for an object infinitely far away from the mirror.

49. Properties of Spherical Mirrors • For any spherical mirror whose size is small compared to its radius of curvature, the focal length is given by: • Where f is the focal length and r is the radius of the mirror.

50. Properties of Spherical Mirrors • One last comment: this analysis is an approximation. • The rays don’t all converge exactly at F. • For more curved mirrors, this can lead to a blurred image.