#### Transcript Reflect/Refract

Optics Reflection and Refraction Reflection •What happens when our wave hits a conductor? •E-field vanishes in a conductor •Let’s say the conductor is at x = 0 •Add a reflected wave going other direction •In reality, all of this is occurring in three dimensions Incident Wave Reflected Wave Total Wave Ei E0 sin kx t Er E0 sin kx t Waves Going at Angles •Up to now, we’ve only considered waves going in the x- or y-direction •We can easily have waves going at angles as well E E sin k x k y t i 0i x y 2 2 ck c kx k y Er E0 r sin k x x k y y t •What will reflected wave look like? •Assume it is reflected at x = 0 c k x2 k y2 •It will have the same angular frequency •Otherwise it won’t match in time k y k y •It will have the same ky value •Otherwise it won’t match at boundary •kx must be negative • So it is going the other way c k x2 k y2 c k x2 k y2 kx2 k y2 kx2 k y2 k x k x k x k x Law of Reflection •Since the frequency of all waves are the same, the total k ki = kr for the incident and reflected wave must be the same. •To match the wave at the boundary, ky must be the same before and after ki sini = kr sinr ki sini sini = sinr kr sinr i r i = r Mirror y x Geometric Optics and the Ray Approximation •The wave calculations we have done assume i = r the mirror is infinitely large •If the wavelength is sufficiently tiny compared to objects, this might be a good approximation i r •For the next week, we will always make this approximation Mirror •It’s called geometric optics •Physical optics will come later •In geometric optics, light waves are represented by rays •You can think of light as if it is made of little particles •In fact, waves and particles act very similarly •First hint of quantum mechanics! Measuring the Speed of Light •Take a source which produces EM waves with a known frequency •Hyperfine emission from 133Cs atom •This frequency is extremely stable •Better than any other method of measuring time •Defined to be frequency f = 9.19263177 GHz •Reflect waves off of mirror ½ ½ •The nodes will be separated by ½ •Then you get c from c = f •Biggest error comes from 133Cs measuring the distance •Since this is the best way to measure distance, we can use this to define the meter •Speed of light is now defined as 2.99792458108 m/s The Speed of Light in Materials •The speed of light in vacuum c is the same for all wavelengths of light, no matter the source or other nature of light c 3.00 108 m/s •Inside materials, however, the speed of light can be different •Materials contain atoms, made of nuclei and electrons Indices of Refraction •The electric field from EM waves push on the electrons Air (STP) 1.0003 •The electrons must move in response Water 1.333 •This generally slows the wave down c Ethyl alcohol 1.361 v •n is called the index of refraction n Glycerin 1.473 •The amount of slowdown can depend Fused Quartz 1.434 on the frequency of the light Glass 1.5 -ish Cubic zirconia 2.20 Diamond 2.419 Refraction: Snell’s Law ck •The relationship between the angular frequency and the wave number k changes inside a medium n •Now imagine light moving from one medium to another •Some light will be reflected, but usually most is refracted •The reflected light again must obey the law of reflection •Once again, the k1sin1 frequencies all match •Once again, the y-component 1 r index n 1 of k must match ck1 ck2 index n2 n2 k1 n1k2 y n1 n2 k1 sin 1 k2 sin 2 x n1n2 k1 sin 1 n2 n1k2 sin 2 n1 sin 1 n2 sin 2 Snell’s Law 2 k2sin2 c f n 1 = r Dispersion •The speed of light in a material can depend on frequency •Index of refraction n depends on frequency •Confusingly, its dependence is often given as a function of wavelength in vacuum •Called dispersion •This means that different types of light bend by different amounts in any given material •For most materials, the index of refraction is higher for short wavelength Red Refracts Rotten Blue Bends Best Prisms •Put a combination of many wavelengths (white light) into a triangular dispersive medium (like glass) •Prisms are rarely used in research •Diffraction gratings work better •Lenses are a lot like prisms •They focus colors unevenly •Blurring called chromatic dispersion •High quality cameras use a combination of lenses to cancel this effect Rainbows •A similar phenomenon occurs when light bounces off of the inside of a spherical rain drop •This causes rainbows •If it bounces twice, you can get a double rainbow Total Internal Reflection •If sin 2 comes out bigger than one, then none of the light is refracted •It is all reflected •This can only happen if it is going from a high index to low index material •The minimum incident angle where this happens is called the critical angle n1 sin c n2 sin c n2 n1 n1 sin 1 n2 sin 2 1 n1 2 n2 Optical Fibers Protective Jacket Low n glass High n glass •Light enters the high index of refraction glass •It totally internally reflects – repeatedly •Power can stay largely undiminished for many kilometers •Used for many applications •Especially high-speed communications – up to 100 Tb/s Fermat’s Principle (1) •Light normally goes in straight lines. Why? •What’s the quickest path between two points P and Q? •How about with mirrors? Go from P to Q but touch the mirror. •How do we make PX + XQ as short as possible? •Draw point Q’, reflected across from Q •XQ = XQ’, so PX + XQ = PX + XQ’ • To minimize PX + XQ’, take a straight line from P to Q’ i = r P We can get: (1) light moves in straight lines, and (2) the law of reflection if we assume light always takes the quickest path between two points Q i r X i Q’ Fermat’s Principle (2) •What about refraction? •What’s the best path from P to Q? •Remember, light slows down in glass •Purple path is bad idea – it doesn’t avoid the slow glass very much •Green path is bad too – it minimizes time in glass, but makes path much longer •Red path – a compromise – is best •To minimize, set derivative = 0 P n1 sin 1 n2 sin 2 1 s1 1 L – x d1 x s1 s2 n1s1 n2 s2 1 2 2 2 t n1 x d1 n2 L x d 22 v1 v2 c c c n L x dt 1 n1 x 2 1 n sin n sin 0 1 1 2 2 2 2 dx c x 2 d12 c L x d2 d2 2 s2 2 Q Light always takes the quickest path