1. Lecture 36 Fermat’s Principle Reflection and refraction
Spherical mirrors
Thin lenses
Magnification

2. Light as wave and particles
Light is a wave, as discovered by Young et al
It has interference
It has diffraction
We will discuss these properties in late part
However, we also know that light travels on a straight-line, just like particles.
According to Newton, light is a beam of particles
Particle-wave duality: light is particle and but also wave

3. Reflection and refraction

4. Laws of reflection and refraction
Angle of incidence = angle of reflection.
𝜃 𝑟 = 𝜃 𝑎
Snell’s Law of Refraction
sin 𝜃 𝑎 sin 𝜃 𝑏 = 𝑛 𝑏 𝑛 𝑎
considers the slowing of light in a medium other than vacuum
𝑐 𝑛 = 𝑐 𝑛
Where n is the index of refraction.

5. Propagation of Light, Fermat’s Principle (1657)
Involves the principle of least time: The path between two points that is taken by a beam of light is the one that is traversed in the least amount of time.
𝑇= 𝑡 0 𝑡 1 𝑑𝑡 = 1 𝑐 𝑡 0 𝑡 1 𝑐 𝑣 𝑑𝑠 𝑑𝑡 𝑑𝑡 = 1 𝑐 𝐴 𝐵 𝑛𝑑𝑠
A more modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path.
𝛿𝑇=𝛿 1 𝑐 𝐴 𝐵 𝑑𝑠 =0

6. Fermat's principle leads to Snell's law
To find the path of least time, set dt/dx=0.
Since ni = c/vi and nt = c/vt . Snell’s Law of Refraction

7. Constructing the image from a plane mirror
Following the light rays to form an image of an object.

8. The focal point and focal length of a spherical mirror
The focal point is at half of the mirror’s radius of curvature.
All incoming rays will converge at the focal point.

9. Thin mirrors
𝑂𝐵𝑐𝑜𝑠𝛼= 𝑅 2 𝑂𝐵= 𝑅 2𝑐𝑜𝑠𝛼 If 𝛼 is small 𝑂𝐵= 𝑅 2 (1+ 𝛼 2 2 ) For thin convex or concave mirrors 𝑂𝐵≈ 𝑅 2 A 𝑅 𝛼 𝛼 O B

10. Convex spherical mirror

11. A system of rays may be constructed to reveal the image
Rays are drawn with regard to the object, the optical axis, the focal point, and the center of curvature to locate the image.

12. Refraction within a sphere
Making use of Fermat’s Principle, one can get when ℎ ≪𝑠′ 𝑎𝑛𝑑 𝑠 𝑛 𝑎 𝑠 + 𝑛 𝑏 𝑠 ′ = 𝑛 𝑏 − 𝑛 𝑎 𝑅 When 𝑠→∞, 𝑠′ is called the focal length 𝑓 ′ =𝑅 𝑛 𝑏 𝑛 𝑏 − 𝑛 𝑎 When 𝑠 ′ →∞, one gets another focal length 𝑓=𝑅 𝑛 𝑎 𝑛 𝑏 − 𝑛 𝑎

13. Thin lenses

14. Thin lenses

15. Converging lenses
1 𝑠 + 1 𝑠 ′ = 1 𝑓

16. Magnifying glass 1 𝑠 + 1 𝑠 ′ = 1 𝑓 Magnification M= 𝑠 ′ 𝑠 = 𝑠 ′ −𝑓 𝑓
optical axis F • 𝑠 𝑠′
1 𝑠 + 1 𝑠 ′ = 1 𝑓
Magnification M= 𝑠 ′ 𝑠 = 𝑠 ′ −𝑓 𝑓
The maximum angular magnification (compared to the naked eye) of a magnifying glass depends on how the glass and the object are held, relative to the eye. If the lens is held at a distance from the object such that its front focal point is on the object being viewed, the relaxed eye (focused to infinity) can view the image with angular magnification
𝑀𝐴= 25𝑐𝑚 𝑓 𝑜𝑟 𝑀𝐴= 25𝑐𝑚 𝑓 +1

17. Diverging lens

18. The camera
A clever arrangement of optics with a method to record the inverted image on its focal plane (sometimes film, sometimes an electronic array, it depends on your camera).