Tuesday, Nov. 9 Chapter 12: Wave Optics We are here Geometric optics compared to wave optics Phase Interference Coherence Huygens principle & diffraction Slits and gratings Diffraction patterns & spectra Thin films Demos: Vibraphone bars, two speaker interference, two slit interference
Two approaches to optics 1. Geometric optics (we did that already) Treats light as rays that are lines (ray tracing) Explains: images, mirrors and lenses Light rays travel in straight lines, and 2 rays always add to brightness. 2. Wave optics Treats light as waves with wave fronts and phase Explains: thin films (oil slicks), holograms, lasers Wave optics is required when objects are similar in size to a wavelength of light. Waves diffract and interfere (two waves can cancel out).
Preview: a difference between ray and wave behavior point source slits (much larger than wavelength) image (shadow projection, as with pinhole camera) Question: how does the image change as the slits move closer together? point source slits (not much larger than wavelength) image (diffraction)?
From Lec. 2: What is a wave? A wave is a propagating disturbance; each disturbance disturbs something next to it, which disturbs something next to it, and so the disturbance travels; although the things of which the disturbance is made may not travel much themselves, the wave travels, and it can carry (transport) energy and momentum. Water waves, sound waves, and string waves (a violin string) are familiar examples of waves, though one must observe carefully to distinguish important wave behaviors. Heavy traffic can exhibit wave behavior; so can crowds in a stadium. Are there other examples?
What does a (sinusoidal) wave look like? wavelength (if horiz. axis is position) or period (if horiz. axis is time; period is 1/frequency) Amplitude (of some physical quantity: for a sound wave, pressure; for a light wave, electric field strength) position or time Wikipedia The wave amplitude varies in both space and time.
What is the wavelength of visible light? compared to 1 meter? compared to (the diameter of) a human hair? compared to a cell? compared to DNA? compared to an atom? Because of the wave nature of light, you can t see (optically distinguish) things smaller than a wavelength.
Phase describes when/where the amplitude is at a minimum/maximum/etc. Amplitude (of some physical quantity: for a sound wave, pressure; for a light wave, electric field strength) Wikipedia =0 phase On this graph, the phase is in radians. π radians = 180 degrees.
Phase How is phase measured? Points on a sine wave can be labeled in degrees like the points around a circle. On full cycle of up and down is 360 degrees. = 0
Wave-fronts are lines of constant phase. Spherical wave fronts are common. Wave fronts from a small point-like source spread out spherically. ray Example: drop a pebble in a pond. Think how phase varies: in time, as you look in the same place; and in space, if you take a snapshot at a given time. Typically we draw the wave-fronts corresponding to the crests (90 deg. phase for a sine wave); half-way in between are the troughs (270 deg. phase for a sine wave).
Interference: in phase or out of phase? Waves add constructively when in-phase: the crests of one wave overlap the crests of the other, and troughs overlap troughs. These waves are in phase: The sum has twice the amplitude. The phase difference of these waves is zero. We will often use sound waves in this discussion, because they are familiar.
In phase or out of phase? Waves add destructively (they cancel) when out of phase: the crests of one wave overlap the troughs of another wave. These waves are out of phase (more carefully, 180 degrees out of phase): The phase difference of these waves is 180 degrees. 0 degrees on one wave is 180 degrees on the other. 90 degrees on one wave is 270 degrees on the other, etc. If the phase difference is 63.5 degrees, for example, the waves add partially.
Coherent and incoherent sources A coherent source emits a single continuous frequency (and single wavelength.) Example: a tuning fork, a laser. An incoherent source emits many frequencies (or one frequency that is broken into waves with unrelated phases) Loudspeaker
Partial coherence is possible Example: hit the same note many times
Phase difference can vary in time (only if the sources have different frequencies) Example: vibraphone bars, 440 Hz and 441 Hz Sound (air pressure) amplitude at speaker time in phase out of phase in phase
(For the rest of this lecture we ll deal with coherent sources, emitting the same frequency.)
Phase difference depends on position The phase of the wave at your location depends on your distance from the source. The waves below are in phase when they start at the speaker locations and have the same wavelength. Equal distances ½ wavelength difference in distance 1 wavelength difference in distance
Phase depends on distance (1) loud In-phase speakers with equal distances to the listener.
Phase depends on distance (2) soft In-phase speakers with distances to the listener different by ½ λ.
Phase depends on distance (3) loud soft
Phase depends on distance (4) soft loud soft loud
Mathematical statement (for coherent sources) If the sources are in phase, the waves amplitudes add if the distances to the sources differ by nλ, where λ is the wavelength and n is an integer (-1, 0, 1, etc.). If the sources are in phase, the wave amplitudes cancel if the distances differ by (n+ ½)λ. If the sources are out of phase, the rules are reversed.
Coherent sources make an interference pattern soft loud soft loud One pattern: Two patterns: Soft Loud Soft Loud Soft Loud Soft
The pattern that two in phase speakers makes is shown below: The speakers are displaced 2 ½ wavelengths vertically, so there is cancellation along a line in the vertical direction and there is addition in the horizontal direction. In the spaces between the loud areas the sound is softer.
Chapter 12: Wave Optics We are here Geometric optics compared to wave optics Phase Interference Coherence Huygens principle & diffraction Slits and gratings Diffraction patterns & spectra Thin films
From Lecture 2 and earlier in this lecture: What is a wave? A wave is a propagating disturbance; each disturbance disturbs something next to it, which disturbs something next to it, and so the disturbance travels; although the things of which the disturbance is made may not travel much themselves, the wave travels, and it can carry (transport) energy and momentum. Water waves, sound waves, and string waves (a violin string) are familiar examples of waves, though one must observe carefully to distinguish important wave behaviors. Heavy traffic can exhibit wave behavior; so can crowds in a stadium. Are there other examples?
Huygens s Principle Principle: When a wavefront encounters a small slit, what happens on the other side can be found by putting a point source at the slit with the same phase as the incident wave. This behavior (so unlike rays) is called diffraction. This is very different from the ray behavior in a pinhole camera! (The pinhole in a pinhole camera is much larger than a wavelength.)
Gratings (rows of slits) Tricky point: Don t be confused (by these pictures) into thinking that the diffraction pattern is just a shadow projection. (1) The number of slits is not the number of bright lines. (2) More closely-spaced slits produce a more widely-spaced line pattern. This is the opposite of shadow projection.
Find wavelength from diffraction pattern Find spacing between bright spots Zero difference in path length gives constructive interference. x bright spot D D-λ One-wavelength difference in path gives constructive interference. bright spot
Find wavelength from diffraction pattern x = distance between slits s = distance between bright spots D = distance between slits and screen λ = wavelength approximately a right angle (when x is very small) λ Find spacing between bright spots Triangles x and s are similar: D x λ λ x = s D λ = x D s bright spot s D-λ D Distance between slit and screen is nearly D. (This drawing exaggerates small lengths.) Distance between bright spots is nearly s (because x is very small). One-wavelength difference in path gives constructive interference. bright spot
Finding the wavelength of light with pantyhose and a laser: Finely woven cloth, such as nylon stockings, is a grating. Shine a laser beam through cloth and you get a dot pattern like the one above. If you know the spacing between threads (X in our formula), the distance between dots (S in our formula) and the distance from the cloth to the wall (D in our formula), you can calculate the wavelength of light. Demo: Wavelength of light. A cloth is used with 20 threads per millimeter. The distance to the board is 2000 mm and the distance between the dots in the pattern is 25 mm. So the formula gives us SX / D = [25 * (1/20) / 2000] mm for the wavelength of the red light from the laser which is about 625 nm. The laser we used has a known wavelength of 630 nm so our experiment was pretty close. 31
Another way to find the wavelength of light from a diffraction pattern Note that the lemon and lime triangles are similar. The ratio of the shortest side to the longest side is the same for both. So, the formula relating D, X, and λ is λ / X = S / D. Wavelength of light: λ = S X / D. [measure all in the same units] 32