1. OPTICS Visible Light: The visible region of the E&M spectrum. Optics: –Study of the behavior of light. –Principles extend to other E&M radiation. –Many principles apply to sound and other waves. Geometrical Optics or “Ray Model” –Describes waves in terms of lines called “rays”. Physical Optics or “Wave Model” –Describes waves in terms of wavefronts and waves. Quantum Optics –Describes E&M radiation in terms of packets or “quanta” or photons.

2. Geometrical Optics Ray Model: –Light travels in straight lines called rays. –Rays point in direction of propagation of wave Medium: –A material through which a wave propagates. –Light can travel through a medium and it can travel through a vacuum. –Sound can only travel through a medium (since it depends on vibrations of atoms). –When a Ray hits an interface between two media it can be reflected, transmitted, and/or absorbed.

3. Reflection Is actually an absorption and re- emission of light (quantum model). Can be explained with use of rays –Incident Ray from one medium hits the surface of another medium at some point. –The Incident and Reflected Rays define a plane. –We draw an imaginary line normal to the surface at the point of incidence and in the plane of the two rays. Law of Reflection: θ i = θ r where the angles of incidence and reflection are measured relative to the surface normal.

4. Specular and Diffuse Reflection Specular Reflection: –light appears to obey the Law of Reflection from a smooth surface. –A bright spot appears on the surface based on the viewer’s orientation (Bright spot where eye is in line with reflected ray). Diffuse Reflection: –Light reflecting off a rough surface is in different directions because the surface normals are in different directions. –Viewer sees an averaged brightness, with no bright spot. –Surface’s appearance is view independent.

5. Refraction When a ray hits an interface between two media and is transmitted –It changes speed upon entering the new medium –The change in speed causes the ray to bend or “refract”. Index of Refraction (n) for a material: –n = c/v > 1 –where c = speed of light in vacuum –and v = speed of light in medium Snell’s Law of Refraction: (again all angles are measured relative to the surface normal.) Or: n 1 sinθ 1 = n 2 sinθ 2

6. Refraction (cont’d) When light travels from one medium into another: –Speed changes –Wavelength changes –Frequency (color) remains the same. –The relationship v=f λ still holds. –Light passing into medium of larger n bends towards normal. –Light passing into medium of smaller n bends away from normal Total Internal Reflection: Beyond a “critical angle” of incidence, the ray will be refracted back into the original medium.

7. Example: Light passing through a Rectangular Plate We have a glass plate with thickness: d=2cm and n g =1.5. Air has n a =1.0. Light reflecting off an object is incident on the plate with an angle θ i =60 degrees. The light enters the plate and emerges out the other side. –Show that the incident ray is parallel to the emergent ray, but displaced laterally. –Find the lateral displacement (δ) of the emergent ray from the incident ray.

8. Images formed by Plane Mirrors

9. Images Formed by Spherical Mirrors Concave Mirror –Reflection takes place on the inner spherical surface. Rays converge. Convex Mirror –Reflection takes place on the outer spherical surface. Rays diverge. Geometric Properties –Principal Axis: straight line perpendicular to the center of the mirror. –C, R = center and radius of Curvature. –F, f = focus and focal length –f = R/2

10. Ray Tracing to Locate Images Rays parallel to principal axis coming from ∞ when reflected, all pass very nearly through the focus (F). To map a point on an object to its image location, we draw three rays: 1.Ray 1 is drawn parallel to the axis and passes through F after reflection. 2.Ray 2 is drawn through F and its reflected ray is then parallel to the axis. 3.Ray 3 is drawn perpendicular to the mirror (passing through C) and its reflection comes back on itself.

11. Image Formed by a Concave Mirror Case 1: Object is further from mirror than focus: –Image formed is REAL (light actually passes through the image point). –Image is inverted.

12. Image Formed by a Concave Mirror (cont’d) Case 2: Object is closer to the mirror than focus: –Image formed is VIRTUAL (appears behind mirror where no light actually falls.) –Image is upright.

13. Dimensions on a Ray Diagram Object Distance (s o ), Image Distance (s i ) –Are (+) if on the reflecting side of the mirror. –Are (-) if behind the mirror. –Mirror Equation: (1/ s o + 1/ s i = 1/f) Object Height (h o ), Image Height (h i ) –h o is always (+) –h i is (+) if upright; (-) if inverted. Magnification: M = h i / h o = - s i / s o How can an object distance be negative? –If it is an image of a real object and is the effective object of a second mirror or lens.

14. Ray Diagram for a Convex Mirror Image is Virtual and Upright.

15. Sign Conventions for Mirrors For use in The Mirror Equation and for Magnification: 1/ s o + 1/ s i = 1/f ; M = h i / h o = - s i / s o –s o is (+) if the object is in front of the mirror. –s o is (-) if the object is in back of the mirror. –s i is (+) if the object is in front of the mirror. –s i is (-) if the object is in back of the mirror. –f and R are (+) for a concave mirror. –f and R are (-) for a convex mirror. –If M is (+), image is upright –If M is (-), image is inverted.

16. Images Formed by Thin Lenses Images are formed by refracting rays. Rays are very nearly focused to a point if thickness of lens is small compared to radius of curvature (R). Converging Lens –Thicker in the middle than at the ends. –Causes incoming parallel rays to converge Diverging Lens –Thinner in the middle than at the ends. –Causes incoming parallel rays to diverge.

17. Ray Tracing to Locate Images (Thin Lenses) Draw the thin lens as a line; incoming rays refract (bend) as they pass the line. To map a point on an object to its image location, we draw three rays: 1.Ray 1 is drawn parallel to the axis and is refracted by the lens to pass through a focal point (F) (far for converging; near for diverging). 2.Ray 2 is drawn towards the very center of the lens and passes straight through (thin lens). 3.Ray 3 is drawn on a line passing through the other focal point (F′) and emerges parallel to the axis.

18. Image Formed by a Converging Lens Case 1: Object is further from lens than focal point: –Image formed is REAL (light actually passes through the image point). –Image is inverted.

19. Image Formed by a Converging Lens (cont’d) Case 2: Object is closer to lens than focal point: –Outgoing rays must be traced backward to find image. –Image formed is VIRTUAL (light only appears to diverge from the image point.) –Image is upright.

20. Image Formed by a Diverging Lens (cont’d) Outgoing rays must be traced backward to find image. Image formed is virtual and upright.

21. The Lens Equation Relates image distance, object distance and focal length (same as mirror equation): 1/ s o + 1/ s i = 1/f ; M = h i / h o = - s i / s o Sign Conventions: –f is (+) for converging lens and (-) for diverging lens. –s o is (+) if it is on the side from which light is coming (true except sometimes in combinations of lenses), otherwise negative. –s i is (+) if it is on the opposite side of lens from where light is coming, otherwise negative. (s i is (+) for real images and (-) for virtual images.) –h i is (+) for an upright image; (-) if inverted

22. Lenses Immersed in a Media Suppose a converging lens is immersed in water. What would be the effect on its focal length? –Refracted rays bend less; the focal length increases. Suppose a converging lens is put in a vacuum instead of in air? –Refracted rays bend slightly more; focal length decreases. Suppose a converging lens is immersed in a medium that has a higher index of refraction than the lens material. –Refracted rays bend away from the normal; lens becomes diverging.

23. Compound Systems of Lenses The image of the real object becomes the “object of the next lens”. This image is called a “virtual object” Example : Image formation by a compound microscopeImage formation by a compound microscope –AB is the actual (real) object seen by the “objective lens”. –A´B´(image of AB) is the virtual object for the “eyepiece lens”. –A´´B´´ is the final enlarged, virtual image.

24. Films at Hippocampus Assignment: View these animations: After clicking Here, at Hippocampus.org Physics APB IIHere, at Hippocampus.org Physics APB II –Scroll down to Geometric Optics –Then click on Flat and Concave Mirrors (view animation). –Then click Lenses (view animation). –Then click on Images (view animation).

25. Physical Optics (Wave Model) How can we predict the path of propagation of a wave? –Wavefront: the entire width of a crest or trough. –For a spherical wave propagating from a point source Wavefronts are spherical surfaces Circular in two dimensions Radius of Curvature increases with distance from source We have Plane Waves when R  ∞ Plane Waves are Forever. Huygens Principle –provides us with a way to graphically construct successive wavefronts from preceding wavefronts.

26. Huygen’s Principle To construct a subsequent wavefront from an existing wavefront: –Every point on an existing wavefront can be considered as a point source of spherical wavelets spreading out at the wave speed. –The new wavefront is the surface that is tangent to these small “wavelet fronts”. (a) Spherical Waves: Circular wavefronts that get flatter. (b) Plane Waves Flat wavefronts that stay flat.

27. How does a Plane Wave propagate through a Small Slit? (length ~ wavelength) Diffraction: Applying Huygens’ Principle Shows that the wave bends around the corners of the slit (and other obstacles). Do Particles Diffract? Plane Waves propagate through the slit as spherical waves.

28. Young’s Double Slit Experiment Light was passed through two closely spaced slits. Revealed the wave nature of light: Diffraction resulting in Interference. Sunlight (plane waves) passes through single slit creating spherical waves. Spherical waves (crests are red, troughs are yellow) pass through double slit and then to the screen. Interference bands (or “fringes”) result; alternating bright and dark regions.

29. Young’s Experiment (cont’d) Consider each wavelet front drawn to be a crest. Midway between each crest is a trough. When a crest from S1 meets a crest from S2, we get a bright band. When a crest from S1 meets a trough from S2 (and vice-versa) we get a dark band.

30. Young’s Experiment (cont’d) Central Maximum (m=0; zeroth fringe) Animation Double Slit Experiment

31. Young’s Experiment (cont’d) How can we predict where on the screen we’ll find bright and dark bands? –Specify a Point P on the screen by θ –Approximation: d << L, then I 1 and I 2 are nearly parallel at slit –I 2 travels an extra distance (δ) to P than I 1 : δ = d sin θ

32. Young’s Experiment (cont’d) δ = d sinθ is called the path difference between the two rays. When would I 1 and I 2 interfere destructively at P? When would I 1 and I 2 interfere constructively at P? When Path difference is an odd number of half wavelengths or d sinθ = (2m+1) (λ/2) When Path difference is an even number of half wavelengths or d sinθ = (2m) (λ/2) Where: m = 0,1,2, …. = the “order” of the fringe.

33. Young’s Experiment (cont’d) Diffraction occurs when d ~ λ; (radio waves can diffract around mountain tops). Example: –Two slits d =.1 mm; L=1.20 m –Light of wavelength λ = 500 nm passes through –How far apart will the bright fringes be? θ L xmxm d d sin θ = 2m (λ /2) (x m /L) = tan θ ~ sin θ ~ θ x m = (λL/d) m θθ Diffraction occurs when d ~ λ; (radio waves can diffract around mountain tops). Example: –Two slits d =.1 mm; L=1.20 m –Light of wavelength λ = 500 nm passes through –How far apart will the bright fringes be? L θ d L θ

34. Young’s Experiment (cont’d) x m = (λL/d) m (positions of bright spots) So: x 0 = (λL/d) (0) = 0 x 1 = (λL/d) (1) = 6 mm x 2 = (λL/d) (2) = 12 mm So fringes are spaced 6 mm apart. What happens if λ increases? –Spacing increases –Redder light is spaced more widely than blue light. –White light: bright white at central max (m=0) THEN DISPERSION; SPECTROSCOPIC APPLICATIONS –What happens to spacing if experiment is done in water?

35. Coherence When two waves have a distinct phase relationship, they are coherent. –Example: Light split from one source, then recombined to form interference fringes. Incoherence: Light sources with random phase differences. These will interfere randomly and produce uniform brightness instead of fringes.

36. The Diffraction Grating When coherent light is passed through a series of slits, the peaks become sharper; more pronounced: –Maxima are located by the same geometry as used for Young’s double slit: sinθ = mλ/d; m=0, 1, 2, 3, … –Edges of wider two-slit peaks are removed by destructive interference by light coming through slits much further away. Top: 2 slit Bottom: 5 slit

37. Diffraction Gratings (cont’d) Types: –Transmission: arrays of slits allowing light to pass through. –Reflective: grooves that reflect light. Ex: CD’s can act as reflective gratings. Applications: –Spectroscopy: Can reveal the electronic structure (energy levels) and composition of a material. –Crystallography: Can reveal the atomic structure (atomic positions) of a material. Spaces between atoms act like slits.

38. Example For a diffraction grating containing N= 10,000 lines/cm calculate the first and second order angles for 400 nm blue light and 700 nm red light. Note: d = (1/N) cm = 10 -4 cm; sinθ = mλ/d –First Order (m=1) Blue: sinθ = mλ/d = (1)(400nm)/ 10 -4 cm =.400; θ=23.6 o Red: sinθ = mλ/d = (1)(700nm)/ 10 -4 cm =.700; θ=44.4 o –Second Order (m=2) Blue: sinθ = mλ/d = (2)(400nm)/ 10 -4 cm =.800; θ=53.1 o Red: sinθ = mλ/d = (2)(700nm)/ 10 -4 cm = 1.40; θ = !! The second order red fringe does not appear.

39. Thin Film Interference When light reflects off a surface of higher index of refraction, it gets a λ/2 phase shift. Find net path difference between incident and reflected waves. Oil Film on WaterReflective Coating on Lens Thickness of film plus λ/2 phase shift cause constructive interference of red light. Thickness of coating plus two λ/2 phase shifts cause destructive interference of green light.

40. Single Slit Interference Instead of treating a slit as a “point emitter” of light, we view a slit as being an array of point emitters. As with the grating, there are mutual constructive and destructive interferences giving maxima and minima. Maximum at θ = 0 o and Minima found at: wsin θ = Nλ; N = 1, 2, 3, ….; N≠0 Not usually quantitatively on AP B exam.

41. Single Slit Applets Single Slit Diffraction Applet: Can vary wavelength and slit width.Single Slit Diffraction Applet: Can vary wavelength and slit width. wsin θ = Nλ; N = 1, 2, 3, ….; N≠0

42. Optics Reflection of Light and Mirrors