Course OPTO 251 Optics

1. Course OPTO 251 (Optics) Dr. Ashraf Eldakrouri

2. Course Grades OPTO 251 2

3. Course Index OPTO 251 3

4. Lecture 1 Nature of Light Light is made-up of photons which are very small particles of energy. When these photons move or the light travels, it travels in straight lines but in small waves with a velocity called a speed of light. The speed of light is approximately 300 000 km per second.  An example would be tossing a pebble into a pond. The ripples produced, (small waves) travel in a straight line away from the source, the pebble.  Light shares the characteristics of both particles and waves. Light is a form of radiant energy that you can detect with your eyes. Light energy comes from chemical energy, electrical energy and nuclear energy. It is a combination of electrical and magnetic energy that travels very, very fast. The light can pass through anything that is transparent, sort of passes through translucent objects (frosted window) but doesn't make it through opaque objects such as a brick wall. OPTO 251 4

5. Types o Light There are two types of light: luminous - objects that emit their own light (sun) and non-luminous - objects that do not emit light (flashlight that is switched off). Examples of luminous light sources are: a) Light from incandescence - the process of emitting light because of high temperature b) Light from electrical discharge - the process of emitting light because of electricity passing through a gas c) Light from fluorescence - the process of emitting light while receiving energy from another source d) Light form phosphorescence - the process of emitting light for some time after receiving energy from another source OPTO 251 5

6. Energy of Light The Scientists called the light as electromagnetic radiation. Visible light is only one small portion of a family of waves called electromagnetic (EM) radiation. The entire spectrum of these EM waves includes radio waves, which have very long wavelengths and both gamma rays and cosmic rays, which are at the other end of the spectrum and have very small wavelengths. Visible light is near the middle of the spectrum. The key thing to remember is that light and EM radiation carry energy. The quantum theory suggests that light consists of very small bundles of energy/particles; it's just that simple. Scientists call those small particles photons, and the wavelength determines the energy and type of EM radiation, and the number of photons tells you how much radiation there is. A lot of photons give a brighter, more intense type of light. Fewer photons give a very dim and less intense light. When you use the dimmer switch on the wall, you are decreasing the number of photons sent from the light bulb. The type of light is the same while the amount has changed. OPTO 251 6

7. The Electromagnetic spectrum The electromagnetic spectrum is all the different wavelengths of electromagnetic radiation including radio waves, light waves and x-rays to name a few. Light waves or visible light covers the range of wavelength from 400 to 700 nm which is the range in size from a molecule to a protozoan. The sun emits most of its radiation in the visible range which we perceive as the colors of the rainbow. Our eyes are sensitive to this small section of the entire electromagnetic spectrum. OPTO 251 7

8. Examples of EM spectrum 1-Cosmic Rays - background radiation - particles of enormous energy given off  by stars 2- Gamma Radiation - deadly high energy given out by the sun and other stars 3- X-rays - high energy used in x-ray equipment 4- Ultraviolet Rays - invisible sunlight energy waves that cause the skin to tan 5- Infrared Rays - rays of heat energy - felt by our nervous systems 6- Radio Waves - microwaves. Radio energy, TVs 7- Visible Light - the basic colors of light emitted by the sun and visible to the eye OPTO 251 8

9. The Waves As mentioned on the first page, if you tossed a pebble into the water the result would be circular ripples moving outward. These ripples are waves, with a series of crests and troughs. The crests are the high points where the electrical field is the highest and the trough, the low points, have the lowest electrical fields. A wavelength is the distance between two troughs or two crests and the frequency is the number of wavelengths that pass through a given point in one second. The speed of the wave equals the frequency times the wavelength. Longer wavelength has less energy that means the higher the frequency the shorter the wavelength and the lower the frequency the longer the wavelength. The amplitude is a measure of how much energy the wave has. Electromagnetic are transverse waves. As the wave moves along, the particles in the substance move either up and down or side to side. After the wave has passed the particles are back to where they started. It is like a cork floating in water. The wave goes by and the cork bobs up and down but is still in the same place. OPTO 251 9

10. Color of the spectrum Color is only possible because of light. Visible light is made up of all different colors. The different colors are caused by the different wavelengths of light. We can see the different colors of light when we see a rainbow. The rain droplets acts like tiny prisms and causes the light to breakup into its different colors. A glass prism will disperse the light rays and a rainbow (spectrum) will be seen. Red, green and blue are the primary colors because when added together will produce white light. By mixing primary colors in pairs, secondary colors are produced. With every color, there are three main characteristics: hue, lightness (brightness) and chrome. OPTO 251 10

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12. Hue: gives colors their name - red, blue, green etc. - all are different hues of color Lightness: (brightness) the amount of light reflected from the color Chrome: refers to the concentration of color. The Wavelength tells you the type of light but Amplitude tells you the intensity of the light. OPTO 251 12

13. Rectilinear propagation of light (light travels in straight lines) Optical medium it is the medium (space) that the light can travel through it. the optical media have the same properties in all direction and it is called isotropic The Wave-front if a light start out from a point source in an isotropic homogeneous medium. The light will spread out uniformly in all direction with the same speed ( Speed of light). The image of the light at any moment will be sphere with the image of the original source point at the center. This type of image with the source at the center is called wavefront. OPTO 251 13

14. Pencils and beams Pencil of light When the light pass from a light source through a limiting a aperture, it will form a small group of rays called Pencil of light (bundle) The aperture that formed the pencil it may be an actual hole in an opaque screen or the edges of lens mirror or window. The light from the pencil it may be Divergent, convergent or parallel Depend on the type of lenses. The convergent case is called the focus point, where the image should be. Behind the focal point the beam back again to divergent case and the focal point in this case become a new divergent pencil of light . OPTO 251 14

15. In the below figure BC’D’ is the section of a pencil diverging from a luminous point B, and limited by an aperture CD at position A. The width of the rectangular aperture is GE and FH and a rectangular patch of light is formed on a screen at position A’. The patch has the height C’D’ and the corners E’, F’, G’ and H’. In the similar triangles ABC and A’ BC C’A’ / CA = BA’ / BA By the same way we get E’F’ / EF = BA’ / BA E’G’ / EG = BA’ / BA etc. From the above equation we have Area E’F’H’G’ / Area EFHG = E’F’ X E’G’ / EF X EG = (BA’)2 / (BA)2 This means that the are of the cross-section of a pencil varies as the square of its distance from the source OPTO 251 15

16. How do we see the world? OPTO 251 16

17. Pinhole camera model Pinhole model: •Captures pencil of rays–all rays through a single point •The point is called Center of Projection (COP) •The image is formed on the Image Plane •Effective focal lengthfis distance from COP to Image Plane OPTO 251 17

18. A pinhole camera it is a very simple camera with no lens and a single very small aperture. Simply explained, it is a light-proof box with a small hole in one side. Light from a scene passes through this single point and projects an inverted image on the opposite side of the box. Cameras using small apertures and the human eye in bright light both act like a pinhole camera. The smaller the hole, the sharper the image, but the dimmer the projected image. Optimally, the size of the aperture should be 1/100 or less of the distance between it and the screen. A pinhole camera's shutter is usually manually operated because of the lengthy exposure times, and consists of a flap of some light-proof material to cover and uncover the pinhole. Typical exposures range from 5 seconds to hours and sometimes days. A common use of the pinhole camera is to capture the movement of the sun over a long period of time. This type of photography is called Solargraphy. The image may be projected onto a translucent screen for real-time viewing (popular for observing solar eclipses; see also camera obscura), or can expose film or a charge coupled device (CCD). Pinhole cameras with CCDs are often used for surveillance because they are difficult to detect. OPTO 251 18

19. Reflection and Mirrors Chapter 2

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23. Spherical mirror In the spherical mirror is working as a curved mirror. As the plane mirror the Incident angle = the reflection angle The different is the normal is changed at every point due to the curvature of the mirror. As a result of curvature the image now is not at the same distance from mirror as the object but it is more complicated Most image will be before or after or at specific point (distance) from the mirror called Focal point The Focal point is control the shape of the image and if it is virtual or real image. The focal point of the Spherical mirror is depend on the Radius of the mirror. OPTO 251 23

24. Concave Mirror OPTO 251 24

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27. Images formed by spherical mirrors Let us study the image formed by using the concave mirror. A- image formed by an object O which is located outside the center of curvature of the mirror. Note that the image is between the focal point F and the center of curvature C. the image is real, inverted and smaller than the object. OPTO 251 27

28. B -the object O is located at the of curvature C .the concave mirror forms an image at the center of curvature which is real, inverted, and the same size C- the object is located between E and F ray tracing shows that the image is located beyond the center of curvature. It is real ,inverted ,and larger than the object OPTO 251 28

29. D-When the object is at the focal point F, all reflected rays are parallel. Since the reflected rays are parallel either direction no image will be formed {some other prefer to say that the image distance is infinite} E- When the object is located inside the focal point F, The image appears to be behind the mirror .this can be seen by extending the reflected rays to a point behind the mirror. The image will be virtual, enlarged and erect (right side up) OPTO 251 29

30. On the other hand, all image formed by convex mirror have the same characteristics. Such image are virtual, erect and reduced in size. Some application like the mirror in Cars rear-view mirrors OPTO 251 30

31. The Mirror Equation Let us consider the image formed by an extended object is OA. The image of the point O is I. by tracing rays from the top of the arrow, we are able to drew the image of A at B. the ray AM passes through the center of curvature and is reflected back on itself. A ray AV which strikes the vertex of the mirror forms equal angles qi and qr, Rays VB and AM cross at B, forming an image of the top of the arrow at that point. The rest of the image IB can be constructed by tracing similar rays for corresponding points on the object OA. The image is real and inverted OPTO 251 31

32. OV = Object distance = P IV = Image distance = q CV = Radius of curvature = R OA = Object size = y IB = Image size = y’ From the figure we have OCA angle and VCM angle is the same amount a OPTO 251 32

33. tan a = y / (p-R) = y’ / (R- q) From which -y’ / y = R-q / p-R 1 But qi =qr so tan qi = tan qr so y/p = -y’ /q 2 From 1 and 2 we got -y / y = q / p = R-q / p-R 3 By rearranging the terms we got 1/p + 1/q = 2/R 4 This relation is know as the mirror equation We can put the relation in focal point form if we know that f = R/2 5 So the equation will be 1/p +1/q = 1/f 6 OPTO 251 33

34. Magnification The magnification M of the mirror is The ratio of the image size to the object size is the. M = Image size / object size = y’ /y 7 The size refers to any linear dimension, e.g., height or width. From equation 3 and the figure mentioned before the magnification will be M = y’ / y = -q / p Where q is the image distance from the mirror p is the object distance from the mirror OPTO 251 34

35. Spherical Aberration Spherical Aberration : it is the focus defect when the ray refracted from the outer edge of the mirror in different focal point then the focal lens Practically, 1- Spherical mirrors from reasonably sharp images as long as their apertures are small compared with their focal lengths. OPTO 251 35

36. A parabolic mirror doesn’t exhibit this defect. 1- Theoretically, parallel light rays incident on a parallel reflector will focus at a single point on the mirror axis. 2- when a small source of light located at the focal point is the principle used in many spotlights and searchlights. 3- the beam emitted from such advice is parallel to the axis of the reflector. OPTO 251 36

37. Examples A source of light 6 cm high is located 60 cm from a concave mirror whose focal length is 20 cm. Find the position, nature and size of the image? Solution 1- the image position q: 1/p +1/q = 1/f 1/q =1/f -1/p = (p –f ) / p f So by inverse the equation q = p f / (p-f) q = 60 x 20 / (60 -20) q= 1200 / 40 = 30 cm q is + so the image is real. The image size y’ will be M = y’ / y = -q / p y’ = - q y / p y’ = - 30 x 6 / 60 = -3 cm the image is inverted Magnitude is M = - y’ / y = -3 / 6 = - 0.5 OPTO 251 37

38. How far should a pencil be held from a convex mirror to form an image one half the size of the pencil? The radius of the mirror is 40 cm. Solution The only information I have is the radius of mirror and magnification M = - q / p = ½ q = - p/2 1 From the mirror equation q = p f / (p – f) 2 Put 1 in 2 we have -p/2 = p f / (p – f) By divide on p we get -1/2 = f / (p – f) 2f = -p + f But f = R/2 So f = - 40/2 = - 20 cm -2 x 20 = -p - 20 p = - 20 +40 = +20 cm OPTO 251 38

39. Refraction Chapter 3

40. Refraction Refraction is the bending of a light ray as it passes obliquely from one medium to another we define qi is the incident of light with the normal to the surface of refractor qr is the angle of refracted light with the normal to the refractor surface . OPTO 251 40

41. Index of refraction and Snell’s Law 1- Light travels slower in a material than in a vacuum but the frequency of light remains the same. 2- refraction takes place because the change in the velocity of light as it passes from one medium to another. 3- The velocity ratio is called the index of refraction or refractive index n: n=c/v. Generally, n>1, but n is often very close to 1 (for air, n=1.0003). where c is the speed of light = 3 x 1010 cm/s or 3 x 108 m/s v is the speed in the refractor medium The Optical Density: It is a property of a transparent material which is a measure of the speed of light through the material OPTO 251 41

42. In two medium we will have two velocity with two refractive index for each medium so V2 / V1= n1 / n2 If n1 is Air so n1= 1 So n2 / n1 = n n is the absolute refractive index of the second medium So in air case V2/V1 = n Where n = real depth / apparent depth Now in two medium case (No air) When light travels from one medium to another part is reflected and part is refracted with direction given by Snell’s Law OPTO 251 42

43. The distance = the velocity x time So The distance AC should be AC= v2t And so BD = v1t sin q2 = v2t / AD sin q1 = v1t So sin q1/ sin q2 = v1 / v2 From the refractive index defination we know n1 = c/ v1 v1 = c / n1 n2 = c/ v2 v2 = c / n2 So sin q1 / sin q2 = n2 / n1 So n1 sin q1 = n2 sin q2 OPTO 251 43

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45. Wavelength and refraction when the light pass from a new medium, the original wavelength will be different then the wavelength inside the medium. i.e. li > lm Where li is the incident wavelength lm is the wavelength inside the medium the relation between the two wavelength (li & lm) As we know that f = c/l so c = f l In air c = f la In medium vm=f lm Where c is the speed of light vm is the speed inside the medium But nm = c /vm nm = f la / f lm 1 OPTO 251 45

46. By divide equation 1 by f we get nm = la / lm 2 Where la = li So lm = la / nm Now by using Snell's law we get But Using 3 in 4 we get OPTO 251 46

47. Dispersion: The dispersion is the separation of light into its component wavelength Apparent Depth The refraction causes an object submerged in a liquid of higher index of refraction to appear closer to the surface then it actually is. By using Snell's law And by using the Apparent Depth q and the Actual depth p in the refractive index n we get OPTO 251 47

48. Lenses and optical instruments Chapter 4

49. Introduction A lens is a transperent object that alters the shape of a wave-front passing through it Lenses are usually constructed of glass and shaped so that refracted light will form images similar to those discussed for mirrors Simple lenses To understand the lens we will use a prism to discuss this properties. From the figure and Snell’s law we notice that the light will bent towards the normal when entering the prism and away from the normal on leaving OPTO 251 49

50. Same if we have two prism base to base. The refractor beam will converge, but it will not came to the focus. In order to focus the light rays to a point , the extremes rays must be deviated more than the central rays. This accomplished by grinding the surfaces. So that they have a uniformly curved cross section. A lens which brings a parallel beam of light to a point focus in this fashion is called a converging lens. OPTO 251 50

51. We have two types of lens Convergence lens Divergence lens Converging lens: It is one which refracts and converges parallel light to a point focus beyond the lens. Structure of the converging lens: 1- It is constructed with two spherical surfaces. 2- The line joining the centers of the two spheres is known as the axis of the lens 3- there are three types of converging lens Double convex Plano convex Converging meniscus 4- the converging lens is thick in the middle then the edge OPTO 251 51

52. 2- divergence lenses It is one which refracts and diverges parallel light from a point located in front of the lens Structure of the diverging lens: 1- It is constructed with two spherical surfaces like the converging lens 2- it is thicker at the edge then the center 3- parallel light rays passes through such a lens bend towards the thicker part, causing the beam to diverge. 3- there are three types of diverging lens Double concave Plano concave diverging meniscus OPTO 251 52

53. Focal length and lens marker’s equation The focal length (f): The focal length of a lens is reckoned as the distance from the optical center of the lens to either focus. As the mirror the image formation by thin lens is depend on the focal lens but it is not the half of curvature radius. The different is in lens the light pass through the two surface so we have two focal lens as in figure. In convex it is real focal F and in concave it is virtual focal F’ OPTO 251 53

54. The equation of lens Because the focal length is depend on the radius of curvatures and the refractive index . And because the lens have two surface so we have two radius of curvature R1 and R2 So 1/f = (n-1) (1/R1 + 1/R2) Notice 1- The radius of curvature consider positive if it is convex 2- The radius of curvature consider negative if it is concave 3- The focal length of converging (convex) is positive 4- The focal length of diverging ( concave) is negative OPTO 251 54

55. Image formation by thin lens The real image is always formed on the opposite side of the object. The virtual image is formed in the same side of the object. 1- the object located beyond twice the focal length. The image will be real, inverted and formed between F2 and 2F2 on the opposite side of the lens. OPTO 251 55

56. 2- object at a distance equal to the twice of the focal length. The image will be real, inverted and the same size located at the 2F2 on the opposite side of the lens. 3- Object at a distance between 2F1 and F1. the image will be real, inverted and enlarge beyond F2 on the opposite side of lens OPTO 251 56

57. 4- Object at the first focal point F1. No image will formed . The refraction ray is parallel. 5- Object located inside the first focal length. The image is virtual, erect. And enlarge on the same side of object. OPTO 251 57

58. In all cases the image formed by the convex lens is the same behavior like the image formed by the concave mirror both working ad convergence . Also the image formed by the concave lens is the same as the image formed by the convex mirror both is divergence. The concave lens image is always virtual, erect and diminished size. OPTO 251 58

59. The lens equation and magnification The characteristics, size and location of the image can also be determined analytically from the lens equation. And as the mirror equation for the focal length and the image size and location we have 1/p + 1/q = 1/f Where p is object distance q is the image distance f is focal length of lens y is the object size y’ is the image size Consider p & q is + for real image And – for the virtual image Also f focal length is positive for convergence And negative for divergence OPTO 251 59

60. So the equation will be p = f q / (q – f) q = f p / (p – f) f = q p / (p + q ) Also the magnitude will be M = y’ / y = - q / p If we have combine between two lens the magnitude will be M1 = y’1 / y1 M2 = y’2 / y2 But y’1 = y2 because the image of first lens is the object of the second lens So M1 x M2 = y’1 / y1 x y’2 /y2 = y’2 / y1 M1 x M2 is called the magnitude of the combine lenses M M = M1 x M2 = y’2 / y1 OPTO 251 60

61. Ophthalmic prism As we now: 1- Ophthalmic lenses are used to correct anomalies of refraction ( myopia, hyperopia and astigmatism). This occurs by changing the vergence of light incident on their front surface. 2- in some cases, the prism is used to correct the binocular anomalies ( such as uncompensated exophoria and hyperphoria) . This occurs by changing the direction of incident light without affecting its vergence. 3- the prism incorporated into the lens by placing its base apex meridian in the desired direction A prism may also be introduced by decentering a lens. This is why it is necessary to measured the Pupillary distance PD accurately so as to avoid inadvertent lens decentration causing the patient not to look through optical center of the lens and therefore experience a disturbing prismatic effect. OPTO 251 61

62. 4- In ophthalmic optics, only thin prisms, those of small apical angle, are important and only rays incident nearly normally are considered. That simplifies the problem of tracing rays through the prism a lot. The diagram below shows such a prism. We want to calculate the deviation, e, of a ray passing through the prism. 5- Let's simplify the calculation by having the ray incident normally on the first face of the prism. It is not, then deviated by the first face and strikes the second face such that it makes an angle a with the normal. It leaves the prism at an angle ?' with the normal. From Snell's law, n sina=n’sinb n a?n’b 1 But the angle of deviation e is related to ?' by b=a+e 2 Eliminating ?' from (1) and (2), e=[(n/n’ )-1]b . But in the prism case n’ =1 because the front face of the prism in air e = [n - 1 ] b OPTO 251 62

63. Prentice’s Rule We can think of a lens as a stack of prisms, each with slightly different power. Not surprisingly, then, extra-axial points behave as if they had a prism power as well as a lens power. We can calculate that prism power from the diagram below. Consider a parallel bundle of rays striking a lens of power F a distance d from the optical axis. Rays converge to the secondary focal point. The ray bundle is rotated through an angle e where e=-d/f'= d F. To get the prismatic deviation, remember that prism power P is one hundred times the angle of deviation in radians so P=100e=100dF. If we give d in centimeters we can drop the factor of 100 and write simply P=d F OPTO 251 63

64. The Sagitta Formula The sagitta formula is a means of specifying the curvature of a surface. h is the chord of the surface C is the center of curvature s is the sagitta ( distance from any point On the circle and the midpoint of the chord r is the radius of the lens curvature The relation between r , s and h math is s = r m (r2 –h2)1/2 (1) Exact formula s = h2 / 2r (2) approximate formula Notice: the approximate formula shouldn’t use with the contact lens From the power formula F = (n-1) / r 3 If we replace r in equation 2 by equation 3 we get s = F h2 / 2(n-1) 4 F = 2(n-1) s / h2 5 OPTO 251 64

65. Sagitta is a measure from the refracting surface to the chord. It is positive or negative depend on the low of optics If we have a lens with two refractive surface this means we have two sagitta s1, s2 and d ( chord diameter) and F1 and F2 the approximate power. So tc –tp = s1 –s2 Where tc is the center thickness tp is the edge thickness S1 and s2 the sagitta of first surface and 2nd surface respectively From approximate power law F = F1 + F2 We get tc - tp = FAh2 / 2(n-1) The following example demonstrates the use of this formula OPTO 251 65