1. Chapter 34
Geometric Optics
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2. Chapter 34
Chapter 33 opener. Of the many optical devices we discuss in this Chapter, the magnifying glass is the simplest. Here it is magnifying part of page 886 of this Chapter, which describes how the magnifying glass works according to the ray model. In this Chapter we examine thin lenses in detail, seeing how to determine image position as a function of object position and the focal length of the lens, based on the ray model of light. We then examine optical devices including film and digital cameras, the human eye, eyeglasses, telescopes, and microscopes.

3. Learning Goals for Chapter 34
How plane mirrors forms image & why concave & convex mirrors form different images.
Plane mirror
No magnification
m = 1
CONCAVE mirror can magnify m >1; also can minimize
CONVEX mirror always minimizes
m < 1
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4. Learning Goals for Chapter 34
How plane mirrors forms image & why concave & convex mirrors form different images.
How images are formed by curved interface between two transparent materials.
What aspects of lens determine types of image produced.
What causes defects in human vision, & how to correct.
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5. Learning Goals for Chapter 34
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6. Learning Goals for Chapter 34
How plane mirrors forms image & why concave & convex mirrors form different images.
How images are formed by curved interface between two transparent materials.
What aspects of lens determine types of image produced.
What causes defects in human vision, & how to correct.
Cameras, Microscopes & Telescopes.
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7. Introduction
Surgeon performing microsurgery needs sharp, magnified view.
Wearing glasses w/ magnifying lenses.
How do magnifying lenses work?
How do lenses and mirrors form images?
Use light rays to understand principles of optical devices such as camera lenses, eyes, microscopes, & telescopes.
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8. Reflection at a plane mirror surface
Light rays from object at point P are reflected from a plane mirror.
Reflected rays entering eye appear as though they had come from image point
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9. Image formation by a plane mirror
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10. Image formation by a plane mirror
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11. Variable Conventions For optics
Symbol
Description
Signs s
Distance from lens/mirror surface to OBJECT
+ on reflecting side of mirror s’
Distance from lens/mirror surface to IMAGE
+ on “real” side of mirror; - on “virtual” side R
Radius of curvature of lens/mirror
+ R = converging lens/mirror
- R = diverging f
focal length of lens/mirror
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12. Variable Conventions For optics
Label
Description
Real Image
Produced by rays that converge
Can be projected onto a screen
Virtual Image
Produced by rays that SEEM to come from an image
Cannot be formed onto a screen.
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13. Image formation by a plane mirror: Sign rules
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14. Characteristics of image from plane mirror
Image is virtual, right-side up, reversed, & same size as object.
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15. Image is reversed by plane mirror
Image of right hand is left!
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16. Formation of Images by Spherical Mirrors
Spherical mirrors are shaped like sections of a sphere, and may be reflective on either the inside (concave) or outside (convex).
Figure Mirrors with convex and concave spherical surfaces. Note that θr = θi for each ray.

17. Image will appear as though it is larger, & apparently much closer…
Concave Spherical Mirrors CONVERGE light beams & form LARGER VIRTUAL images
Image will appear as though it is larger, & apparently much closer…
Figure Mirrors with convex and concave spherical surfaces. Note that θr = θi for each ray.

19. CONVEX Spherical mirrors DIVERGE light beams, and form SMALLER, VIRTUAL images
Image will appear as though it is smaller, and apparently farther away…
Figure Mirrors with convex and concave spherical surfaces. Note that θr = θi for each ray.

22. Formation of Images by Spherical Mirrors
Rays coming from faraway object are effectively parallel.
Figure If the object’s distance is large compared to the size of the mirror (or lens), the rays are nearly parallel. They are parallel for an object at infinity (∞).

23. Formation of Images by Spherical Mirrors
EVERY point on an object is a source of an infinite number of rays going in all directions!
Figure If the object’s distance is large compared to the size of the mirror (or lens), the rays are nearly parallel. They are parallel for an object at infinity (∞).

24. Formation of Images by Spherical Mirrors
EVERY point on an object is a source of an infinite number of rays going in all directions!
Figure If the object’s distance is large compared to the size of the mirror (or lens), the rays are nearly parallel. They are parallel for an object at infinity (∞).
Bottom of tree reflected here from ALL points on Mirror!

25. Formation of Images by Spherical Mirrors
EVERY point on an object is a source of an infinite number of rays going in all directions!
Figure If the object’s distance is large compared to the size of the mirror (or lens), the rays are nearly parallel. They are parallel for an object at infinity (∞).
Top of tree reflected here from ALL points on Mirror!

26. Formation of Images by Spherical Mirrors
Parallel rays striking a spherical mirror do not all converge at exactly same place if curvature of mirror is large:
spherical aberration
Figure Parallel rays striking a concave spherical mirror do not intersect (or focus) at precisely a single point. (This “defect” is referred to as “spherical aberration.”)
Avoided by using parabolic reflector; more difficult & expensive to make
Used only when necessary e.g. in research telescopes.

27. Formation of Images by Spherical Mirrors
If curvature is small, focus is much more precise
focal point is where rays converge.
Figure Rays parallel to the principal axis of a concave spherical mirror come to a focus at F, the focal point, as long as the mirror is small in width as compared to its radius of curvature, r, so that the rays are “paraxial”—that is, make only small angles with the horizontal axis.

28. Formation of Images by Spherical Mirrors
We use ray diagrams to determine image location.
For mirrors, use three key rays, all begin on object:

29. Three key rays for spherical mirror images
Ray parallel to axis.
After reflection it passes through focal point.

30. Three key rays for spherical mirror images
Ray through focal point; After reflection parallel to axis.

31. Three key rays for spherical mirror images
Ray perpendicular to mirror; Reflects back on itself through CENTER of CURVATURE

32. Formation of Images by Spherical Mirrors
For mirrors, use three key rays, all begin on object:
Ray parallel to axis; reflects through focal point.
Ray through focal point; reflects parallel to axis.
Ray perpendicular to mirror; reflects back on itself.

33. Formation of Images by Spherical Mirrors
CHECK with another Ray!!
Ray striking vertex on center line of mirror; Reflects back at same incident angle on the other side of the center line.

34. Formation of Images by Spherical Mirrors
Intersection of rays from a point on the object gives position of image (of that point on the object.)
Get full image using other points

35. Graphical method of locating images
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36. Graphical method of locating images
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37. Formation of Images by Spherical Mirrors
Cases:
Objects farther than Center of Curvature
Objects at Center of Curvature
Objects farther than focal length, but less than Center of Curvature
Objects at focal length
Objects closer than focal length

38. Object Farther than Center of Curvature?
CoC
focus

39. Object Farther than Center of Curvature?
CoC
focus
Reduced Inverted, REAL image

40. Object at Center of Curvature?
CoC
focus

41. Object At Center of Curvature?
Hard to figure which way ray through CoC goes?
CoC
focus

42. Object At Center of Curvature?
Use that 4th ray through vertex of mirror
CoC
focus
Same height Inverted, REAL image

43. Object Between Focus & Center of Curvature?
CoC
focus

44. Object Between Focus & Center of Curvature?
Extend the mirror as IF it was there…
CoC
focus

45. Object Between Focus & Center of Curvature?
Use that 4th ray to help!
CoC
focus
Magnified Inverted, REAL image

46. Object at Focus?
focus
CoC

47. Object Closer than Focus?
CoC
focus
Magnified, UPRIGHT, VIRTUAL image

48. Object at Focus?
CoC
focus
NO convergent image!!!

49. Object Closer than Focus?
CoC
focus

50. Object Closer than Focus?
CoC
focus
Magnified, UPRIGHT, VIRTUAL image

51. Image of a point object at a spherical surface
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52. The Mirror Equation!
Geometrically, derive equation relating object distance, image distance, & focal length of the mirror
1 𝑠 𝑠 = 1 𝑓
Figure Diagram for deriving the mirror equation. For the derivation, we assume the mirror size is small compared to its radius of curvature. s’ s

53. Formation of Images by Spherical Mirrors
Magnification (ratio of image height to object height):
𝑚= 𝑦 𝑖𝑚𝑎𝑔𝑒 𝑦 𝑜𝑏𝑗𝑒𝑐𝑡 = − 𝑠 𝑠
Negative sign indicates that image is inverted.
If object between center of curvature and focal point, image is larger, inverted, real.

54. Example: Image in a concave mirror.
A 1.50-cm-high diamond ring is placed 20.0 cm from a concave mirror with radius of curvature 30.0 cm. Determine (a) the position of the image, and (b) its size.
Solution: a. Using the mirror equation, we find di = 60.0 cm.
b. Using the magnification equation, we find M = and hi = -4.5 cm.

55. Formation of Images by Spherical Mirrors
If object is outside center of curvature of a concave mirror, image will be inverted, smaller, & real.
Figure You can see a clear inverted image of your face when you are beyond C (do > 2f), because the rays that arrive at your eye are diverging. Standard rays 2 and 3 are shown leaving point O on your nose. Ray 2 (and other nearby rays) enters your eye. Notice that rays are diverging as they move to the left of image point I.

56. Example: Object closer to concave mirror.
A 1.00-cm-high object placed 10.0 cm from concave mirror w/ radius of curvature = cm.
(a) Draw a ray diagram to locate (approximately) the position of the image. (b) Determine the position of the image and the magnification analytically.
Figure Object placed within the focal point F. The image is behind the mirror and is virtual, [Note that the vertical scale (height of object = 1.0 cm) is different from the horizontal (OA = 10.0 cm) for ease of drawing, and reduces the precision of the drawing.]
Example 32–6. Solution: a. The figure shows the ray diagram and the image; the image is upright, larger in size than the object, and virtual.
b. Using the mirror equation gives di = cm. Using the magnification equation gives M =

57. Object closer to concave mirror.
A 1.00-cm-high object placed 10.0 cm from concave mirror w/ radius of curvature = 30.0 cm.
Figure Object placed within the focal point F. The image is behind the mirror and is virtual, [Note that the vertical scale (height of object = 1.0 cm) is different from the horizontal (OA = 10.0 cm) for ease of drawing, and reduces the precision of the drawing.]
Example 32–6. Solution: a. The figure shows the ray diagram and the image; the image is upright, larger in size than the object, and virtual.
b. Using the mirror equation gives di = cm. Using the magnification equation gives M =

58. Spherical mirror image magnification
Ray 1: Angle of incidence = Angle of Reflection!
Ray 2: Light passing through “C” hits mirror normal
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59. Problem Solving: Spherical Mirrors
Draw ray diagram; rays intersect at image.
Apply mirror & magnification equations.
Sign conventions: if object, image, or focal point is on reflective side of mirror, distance is positive (otherwise distance(s) is/are negative.
Magnification is positive if image is upright, negative otherwise.
Check that your solution agrees with the ray diagram.

60. Spherical Mirror Images & Magnification
Equation
Description
1 𝑠 𝑠 = 2 𝑅
Object-image distance relationship for spherical mirrors (only!)
f = 𝑅 2
focal length of spherical mirror
1 𝑠 𝑠 = 1 𝑓
Common Equation for determining image, object, and/or focal lengths for spherical mirrors
𝑚=− 𝑠 𝑠
Magnification (note sign!)
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61. Formation of Images by Convex Mirrors
Figure Convex mirror: (a) the focal point is at F, behind the mirror; (b) the image I of the object at O is virtual, upright, and smaller than the object.
Convex mirrors will not converge incoming light to a focus!
Produce VIRTUAL images only!

62. Image formation by a convex mirror
Convex mirrors have negative R <0
Resulting image is virtual
Image is on opposite side of mirror from object (s’ < 0)
Right-side up! (m > 0)
Smaller than object. (m<1)
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63. Image formation by a convex mirror
Convex mirrors have negative R
Resulting image is virtual
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64. Example: Convex rearview mirror.
External rearview car mirror is convex with radius of curvature of 16.0 m. Determine location of image & magnification for object 10.0 m from mirror.
Solution: The ray diagram for a convex lens appears in Figure 32-19b. A convex mirror has a negative focal length, giving di = -4.4 m and M = The image is virtual, upright, and smaller than the object.

65. Example: Convex rearview mirror.
External rearview car mirror is convex with radius of curvature of 16.0 m. Determine location of image & magnification for object 10.0 m from mirror.
Convex mirror:
Image will be VIRTUAL (behind the mirror)
Radius of curvature = NEGATIVE = -16 m
Object distance “s” = 10 m
Solution: The ray diagram for a convex lens appears in Figure 32-19b. A convex mirror has a negative focal length, giving di = -4.4 m and M = The image is virtual, upright, and smaller than the object.

66. Example: Convex rearview mirror.
External rearview car mirror is convex with radius of curvature of 16.0 m. Determine location of image & magnification for object 10.0 m from mirror.
1 𝑠 𝑠 = 2 𝑅
Solution: The ray diagram for a convex lens appears in Figure 32-19b. A convex mirror has a negative focal length, giving di = -4.4 m and M = The image is virtual, upright, and smaller than the object.
1 𝑠 = 2 𝑅 − 1 𝑠 = 2 (−16) − 1 10 =− .225
so… s’ = m
𝑚= − 𝑠 𝑠 = =0.44

67. What if not in air…?
Must use indices of refraction for BOTH materials to determine image point P’
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68. Image of a point object at a spherical surface
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69. Apparent depth of a swimming pool
IF surface is not curved, but a plane, with R = ∞
How deep are swimming pools?
Deeper than they appear!
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70. Apparent depth of a swimming pool
Object at bottom of pool at actually at depth s
Behind surface! s < 0 !
Image of that object appears at depth s’
Apparent depth of pool is less than its actual depth.
1.33 𝑠 𝑠 =0 => 𝑠’ = 𝑠
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71. Apparent depth of a swimming pool
What about magnification?
m = − 𝑠 𝑠 < 1???
NO!
For PLANE surface, image has same lateral size (m = 1)
Image is upright.
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72. Thin Lenses & Ray Tracing
Thin lenses have thickness small compared to radius of curvature.
They may be either converging or diverging
Figure (a) Converging lenses and (b) diverging lenses, shown in cross section. Converging lenses are thicker in the center whereas diverging lenses are thinner in the center. (c) Photo of a converging lens (on the left) and a diverging lens (right). (d) Converging lenses (above), and diverging lenses (below), lying flat, and raised off the paper to form images.

73. Thin Lenses; Ray Tracing
Parallel rays are brought to a focus by a converging lens
Note! Converging lens will be thicker in center than at edge!
Figure Parallel rays are brought to a focus by a converging thin lens.

74. Types of lenses
Any lens thicker at center than at edges is converging with positive f
Lens thicker at edges than center is diverging lens with negative f.
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75. Thin lens systems
Lens = optical “system” with two refracting surfaces.
Simple lens has two spherical surfaces of same R close together
Could have different R values!
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76. Lensmaker’s equation
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77. Lensmaker’s equation
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78. Lensmaker’s equation
A convex meniscus lens is made from glass with n = The radius of curvature of the convex surface is 22.4 cm and that of the concave surface is 46.2 cm.
What is the focal length?
Where will the image be for an object 2.00 m away?
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79. Thin converging (convex) lens
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80. Thin converging (symmetric bi-convex) lens
Rays passing through first focal point F1 emerge from converging lens as parallel beam of rays.
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81. Thin Lenses; Ray Tracing
Diverging lens is thicker at edge than in center
Makes parallel light diverge;
focal point is where diverging rays would converge (f < 0 negative!)
Figure Diverging lens.

82. Lens Power & Diopters Power of a lens is inverse of focal length:
Lens power is measured in diopters, D (m-1)
Lens f = 1 ⁄ 3 meters (33 cm), D = 3 diopters
D= +10 diopter lens, f = 10 cm (closer!) (good magnifying glass!)

83. Lens Power & Diopters
Power of a lens is inverse of focal length:

84. Lens Power & Diopters Power of lens is inverse of focal length:
Typical human eye without refractive error has power of ~60 diopters (f = 1.7cm)
Eyeglass lenses usually much weaker.
Eyeglasses work by correcting focus, not magnifying
Wikipedia contributors. Eyeglass prescription [Internet]. Wikipedia, The Free Encyclopedia; 2017 Sep 9, 16:34 UTC [cited 2017 Oct 11
]. Available from:

85. Lens Power & Diopters Distant Vision Near Vision
Wikipedia contributors. Eyeglass prescription [Internet]. Wikipedia, The Free Encyclopedia; 2017 Sep 9, 16:34 UTC [cited 2017 Oct 11
]. Available from:

86. Lens Power & Diopters Right Eye Left Eye
Wikipedia contributors. Eyeglass prescription [Internet]. Wikipedia, The Free Encyclopedia; 2017 Sep 9, 16:34 UTC [cited 2017 Oct 11
]. Available from:

87. Lens Power & Diopters Initial Spherical Lens Power (diopters)
Negative = Diverging
Wikipedia contributors. Eyeglass prescription [Internet]. Wikipedia, The Free Encyclopedia; 2017 Sep 9, 16:34 UTC [cited 2017 Oct 11
]. Available from:

88. Lens Power & Diopters Initial Spherical Lens Power (diopters)
Positive = Converging
Wikipedia contributors. Eyeglass prescription [Internet]. Wikipedia, The Free Encyclopedia; 2017 Sep 9, 16:34 UTC [cited 2017 Oct 11
]. Available from:

89. Lens Power & Diopters Secondary Cylindrical Lens Power (diopters)
Correcting Astigmatism
Wikipedia contributors. Eyeglass prescription [Internet]. Wikipedia, The Free Encyclopedia; 2017 Sep 9, 16:34 UTC [cited 2017 Oct 11
]. Available from:

90. Simulation Tools for lens ray tracing
Dubson, M. et. al. (2017) Geometric Optics. Physics Education Tutorials(PhET), University of Colorado.
Henderson, T. (2017) Optics Bench Interactive. The Physics Classroom
Tu, Rick. (2017) Ray Optics Simulation.
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91. Graphical methods for lenses
Shown below is the method for drawing the three principal rays for a real image formed by a converging lens.
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92. Graphical methods for a diverging lens
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93. Thin Lenses; Ray Tracing
Ray tracing for thin lenses is similar to mirrors. We have three key rays:
This ray comes in parallel to the axis and exits through the focal point.

94. Thin Lenses; Ray Tracing
We have three key rays:
This ray comes in parallel to the axis and exits through the focal point.

95. 33-1 Thin Lenses; Ray Tracing
We have three key rays:
This ray comes in parallel to the axis and exits through the focal point.
This ray comes in through the focal point and exits parallel to the axis.

96. 33-1 Thin Lenses; Ray Tracing
We have three key rays:
Parallel to axis and exits through focal point.
Through focal point and exits parallel to axis.
This ray goes through the center of the lens and is undeflected.

97. Thin Lenses; Ray Tracing
Figure Finding the image by ray tracing for a converging lens. Rays are shown leaving one point on the object (an arrow). Shown are the three most useful rays, leaving the tip of the object, for determining where the image of that point is formed.

98. Conceptual Ex. 33-1: Half-blocked lens.
What happens to the image of an object if the top half of a lens is covered by a piece of cardboard?
Solution: The image is unchanged (follow the rays); only its brightness is diminished, as some of the light is blocked.

99. Thin Lenses; Ray Tracing
For diverging lens, use same three rays
Image is upright and virtual.
Figure Finding the image by ray tracing for a diverging lens.

100. Thin Lens Equation; Magnification
Thin lens equation similar to mirror equation:
Figure Deriving the lens equation for a converging lens.

101. Thin Lens: Sign Conventions
focal length f positive for converging lenses (f negative for diverging).
Object distance d is positive when object on same side as light entering lens, otherwise d = negative.
Image distance d’ is positive if image on opposite side from light entering lens; otherwise d’ is negative.
Height of image is positive if image is upright; negative otherwise.

102. Thin Lens: Sign Conventions
focal length f positive for converging lenses (f negative for diverging).
f >0
f < 0

103. Thin Lens: Sign Conventions
2. Object distance d is positive when object on same side as light entering lens, otherwise d = negative.
d >0
Light coming from here da
db < 0

104. Thin Lens: Sign Conventions
3. Image distance d’ is positive if image on opposite side from light entering lens; otherwise s’ is negative.
d’ >0
Light coming from here
d’ < 0

105. Thin Lens: Sign Conventions
4. Height of image is positive if image is upright; negative otherwise.
hi < 0

106. Thin Lens Equation; Magnification
Magnification also same as for mirror:

107. Image formed by a thin converging lens
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108. Lensmaker’s equation
A convex meniscus lens is made from glass with n = The radius of curvature of the convex surface is 22.4 cm and that of the concave surface is 46.2 cm.
What is the focal length?
Where will the image be for an object 2.00 m away?
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109. Lensmaker’s equation n = 1.50 R of convex surface = 22.4 cm
R of concave surface = cm.
What is the focal length?
Where will the image be for an object 2.00 m away?
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110. Lensmaker’s equation n = 1.50 R of first surface = 22.4 cm
R of second surface = cm.
What is the focal length?
Where will the image be for an object 2.00 m away?
f = 30.2 cm and s’ = 35.6 cm
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111. Cameras
When a camera is in proper focus, position of electronic sensor coincides with position of real image formed by lens.
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112. Camera lens basics
focal length f of camera lens is distance from lens to image when object is infinitely far away.
Effective area of lens controlled by adjustable lens aperture (diaphragm): nearly circular hole of diameter D.
Express light-gathering capability of lens in terms of f-number of lens:
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113. The eye
The optical behavior of the eye is similar to that of a camera.
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114. Defects of vision
A normal eye forms an image on the retina of an object at infinity when the eye is relaxed.
In the myopic (nearsighted) eye, the eyeball is too long from front to back in comparison with the radius of curvature of the cornea, and rays from an object at infinity are focused in front of the retina.
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115. Nearsighted correction
The far point of a certain myopic eye is 50 cm in front of the eye.
When a diverging lens of focal length f = −48 cm is worn 2 cm in front of the eye, it creates a virtual image at 50 cm that permits the wearer to see clearly.
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116. Farsighted correction
A converging lens can be used to create an image far enough away from the hyperopic eye at a point where the wearer can see it clearly.
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117. Angular size
The maximum angular size of an object viewed at a comfortable distance is the angle it subtends at a distance of 25 cm.
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118. The magnifier The angular magnification of a simple magnifier is:
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119. The compound microscope
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120. The astronomical telescope
The figure below shows the optical system of an astronomical refracting telescope.
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121. The reflecting telescope
The Gemini North telescope uses an objective mirror 8 meters in diameter.
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