Light Reflectors and Optical Resonators

Transcription

1 Light Reflectors and Optical Resonators Outline Review of Wave Reflection Reflection and Interference Fiber for Telecommunications Optical Resonators 1

2 Reflection of a Normally Incident EM Wave from a Perfect Conductor E i =ˆxE ( H i =ˆ y η Reflected wave e jkz oe jkz E o o ) e jkz E H H E Incident wave e jkz x E z Standing wave pattern of the E-field kz = 2π kz = π H 2E o 0 z Image by Theogeo flickr.com/photos/theogeo/ / on flickr Standing wave pattern of the H-field 2 kz = 3π/2 kz = π/2 2(E o /η o ) z

3 Reflection & Transmission of EM Waves at Boundaries E 1 = E i + E r E 2 = E t H 1 = H i + H r Medium 1 Medium 2 H 2 = H r 3

4 Reflection of EM Waves at Boundaries E 1 (z =0)=E 2 (z =0) Eo i + E r o = E t o H 1 (z =0)=H 2 (z =0) E r μ Eo i E r o E t η = o η = 1 η 1 η2 ɛ r = o η 2 η 1 E = t Eo i t = η 2 + η o 2η 2 = 1 Eo i η 2 + η 1 REFLECTION COEFFICIENT (note that sign of r depends on the relative values of η 2 and η 1 ) TRANSMISION COEFFICIENT 4

5 x MEDIUM 1 MEDIUM 2 Examples of Light Reflection z The im age below indicates the standing wave patterns ( E i + E r ) resulting when an incident wave in medium 1 with amplitude equal to 1 V/m is incident on an interface. Label the graphs (a)-(g) to match them with the description detailed below. c e 1.6 E y,total a) Plot of with medium 1 being air, medium 2, n 2 =2. Normal incidence. E y,total b) Plot of with medium 1 having n 1 = 2, medium 2 being air. Normal incidence. E y,total c) Plot of with medium 1 being air, medium 2 being a perfect conductor b a f g H y,total d) Plot of with medium 1 being air, medium 2 being a perfect conductor E y,total e) Plot of with medium 1 having n 1 =2, medium 2 being air. Angle of incidence greater than critical angle. E y,total f) Plot of with angle of incidence equal to the Brewster angle. 2.0 d 2.0 E y,total g) Plot of with angle of incidence equal to the Brewster angle. Medium 2 is air. 5

6 Remote Sensing of the Environment using radar E 0 E 5 ε = εo μ = μ o σ =0 1 T 12 E 1 E 4 d 2 ε =3.5ε o μ = μ o σ =10 6 S/m E 2 E 3 T 21 EXAMPLE: MEASUREMENT OF THICKNESS OF POLAR ICE CAPS 6

7 Reflectometry measurement of distance to a target by identifying the nodes in the standing wave pattern transmitter receiver Conducting surface d 1 d 7

8 Today s Culture Moment Ice is more reflective than water 70-80% of sunlight reflected by snow 10% reflected by ocean water 20% reflected by vegetation and dark soil 8

9 The Greenhouse Effect Sunlight is reradiated as heat and trapped by greenhouse gasses such as carbon dioxide. Too much carbon dioxide, however, causes the planet to heat up more than usual. Celsius F Future Temperatures? F Year 9

10 Deploy Aluminum Rafts over Dead Ocean Areas? Net excess energy input into planet Earth 1.6 W/m 2. Illuminance on ground level is ~1000 W/m 2 We need to reflect 1.6/1000 of energy back to Balance the Energy IN/OUT Oceans are 90% absorptive (10% reflective) Aluminum is 88% reflective on the shiny side and 80% reflective on the dull side. (Frosted silica might also be able to be used as a reflector) How much of ocean area do we need to cover with 80% reflective sheets of aluminum to balance the energy IN/OUT? (1.6/1000) / (80% - 10%) = 0.23% (of the Earth s surface area) Earth surface area = 5100 million km 2 We need to cover = 1.2 million km 2 equivalent to ~100 years of today s Aluminum production (assuming 50 μm thick Al foil) Dead ocean zones = 0.24 million km 2 Ice fields: North Pole = 9 to 12 million km² Greenland ice sheet = 1.7 million km² South Pole = 14 million km² Image is in the public domain 10

11 METAL REFLECTION Three Ways to Make a Mirror Kyle Hounsell. All rights reserved. This content is excluded from our Creative Commons license. For more information, see MULTILAYER REFLECTION Image is in the public domain Image is in the public domain: File:Dielectric_laser_mirror_from_a_dye_las er.jpg TOTAL INTERNAL REFLECTION 11

12 Dielectric Mirrors can be >99% reflective n hi n lo Simple dielectric mirrors consist of stacked of layers of high and low refractive index. The layers are chosen such the path-length differences of reflections from low to high index layers are integer multiples of wavelengths. Similarly, reflections from low-index layers have path length difference of half a wavelength, but add constructively because of 180 degree phase shift from the reflection. For normal incidence, these optimized thicknesses are a quarter of a wavelength d 1 d 2 1 b 2 d 1 = λ/(4nhi) n N LO R = b = Π i=0 1+ b n d 2 = λ/(4nlo) HI Thin layers with a high refractive index n HI are interleaved with thicker layers with a lower refractive index n LO. The path lengths l A and l B differ by exactly one wavelength, which leads to constructive interference. Source: wikipedia.com 12

13 Opals are an example of dielectric mirrors Colors with λ =2dsin(α) have constructive interference Image is in the public domain Precious opal consists of spheres of silica of fairly regular size, packed into close-packed planes that are stacked together with characteristic dimensions of several hundred nm. λ Silica spheres α d Unique Opals. All rights reserved. This content is excluded from our Creative Commons license. For more information, see 13

14 Fiber to the Home 2.4 Gbps shared by up to 128 users Small Businesses Mbps service rates 2.4 Gbps out 1.2 Gbps in 1 OLT 2 Splitter km reach Splitter Splitter New Buried Development Splitter Time Division Multiplexing 14 Image by Dan Tentler photos/vissago/ / on flickr

15 Fiber to the Home Time Division Multiplexing 2.4 Gbps shared by up to 128 users Small Businesses Mbps service rates Splitter Splitter Gbps out 1.2 Gbps in 1 2 OLT (located in Verizon's central switching office) km reach 1 4 Splitter Splitter An (Optical Network Terminal) is a media converter that is installed by Verizon either outside or inside your premises, during FiOS installation. The converts fiber-optic light signals to copper/electric signals. Three wavelengths of light are used between the and the OLT (Optical Line Terminal): λ = 1310 nm voice/data transmit λ = 1490 nm voice/data receive λ = 1550 nm video receive Each is capable of delivering: Multiple POTS (plain old telephone service) lines, Internet data, Video 15

16 Fiber to the Home Image by uuzinger / on flickr Image by uuzinger / on flickr Bandwidths & Services POTS Upstream Downstream 1310 nm 1490 nm Voice and Voice and Data Data 1550 nm Video Data Video Channels downstream to each home λ = 1490 and λ = 1550 nm Channel upstream from each home λ = 1310 nm Power & Battery Image of by Josh Bancroft on flickr 16

17 Passive Optical Network Optical (PON) Assembly TO Can Laser Single mode fiber TO Can PINS Ball lens Channels downstream to each home λ = 1490 and λ = 1550 nm λ Channel upstream from each home λ = 1310 nm Thin film filters Transmission Upstream Downstream Wavelength (nm) 17

18 Separating Wavelengths Dispersion Image by Ian Mackenzie madmack/ / on flickr Diffraction Sunlight diffracted through a 20 μm slit Image by wonker / on flickr 18 Image is in the public domain

19 Resonators STANDING WAVE RESONATORS H E E o E H z kz = 2π kz = π z Terminate the standing wave with a second wall to form a resonator 19

20 Thin Film Interference E i Image by Yoko Nekonomania flickr.com/photos/nekonomania/ / on flickr 3 t 12 (r 21 ) t 21 E i e jβ 22L t 12 r 21 t 21 E i e jβ 22L r 12 E i E i t 12E i t12r 21 r 21 t 21 E i e t 12 t 21 E i e jβ 22L jβ22l 20

21 Optical Resonator E i E t = [ t 12 t 21e jβ 2L + t 12 e jβ 2L r 21e jβ 2L jβ 2 L r 21 e t ] Ei [ ( jβl 2 = t 12 t 21 e 1+r 12 r 21 e 2jβL + ( r r e 2jβL ) )]... t 12 t 21 e jβ 2L = E 1 r 12 r 21 e 2jβL i E i 21

22 Fabry-Perot Resonance t 12 t 21 e jkl t = 1 r12 r 21 e 2jkL Transmission t Wavelength [μm] max{e 2jk 2L } =1 Fabry-Perot Resonance: maximum transmission min{e 2jk 2L } = 1 minimum transmission 22

23 Total Internal Reflection Beyond the critical angle, θ c, a ray within the higher index medium cannot escape at shallower angles n 2 sinθ 2 = n 1 sinθ 1 θ c = sin 1 (n 1 /n 2 ) TOTAL INTERNAL REFLECTION For glass, the critical internal angle is 42 For water, it is 49 Image is in the public domain INCOMING RAY HUGS SURFACE n 1 = n 2 =1.5 23

24 Waveguide Transport Light Between Mirrors Metal waveguides Dielectric waveguides So what kind of waveguide are the optical fibers? Image by Dan Tentler photos/vissago/ / on flickr 24

25 Fabry-Perot Modes Constructive Interference r 21 = n 2 n 1 n 2 + n 1 Standing Wave E-field λ o1 = 2n 1L m E r = re i ( ) 1 1 Δλ o =2n L m m n 2 L = m(m +1) λ o2 = 2n 2L m +1 λ o =1μm, n 2 =35,. L = 300μm λ 1 =1μm m = 2100 Δλ =5Å 25

26 Plane Waves in Lossy Materials E y =Re{ A 1 e j(ωtkz) } +Re{ A 2 e j(ωtkz) } Ey(z,t) =A 1 e α/ 2z cos(ωt kz)+a 2 e +α/2z cos(ωt + kz) Field strength [a.u.] e ( α/2)z α =2κβ = 2κω c = 4πκ λ Propagation distance [cm] 26

27 Resonators with Internal Loss Image is in the public domain r = ñ 1 ñ 1 ñ n 2ñ1 t = ñ1 + 2 n E t jkl jk r L αl t 1t 2 e t 1 t 2 e e = = Ei 1 r 1r 2e 2j kl 1 r 1r 2e 2jk rl e 2αL...the EM wave loss is what heats the water inside the food 27

28 Laser Using Fabre-Perot Cavity Resonant modes Gain profile Image is in the public domain 28

29 Resonators with Internal Gain What if it was possible to make a material with negative absorption so the field grew in magnitude as it passed through a material? Field strength [a.u.] E(z) = E i e gz e gz Propagation distance [cm] Resonance: Et t1t 2 e jkl t 1t 2e jkrl e αl = = E 2jkL i 1 r e 2jkL 1r 2 1 r 1r 2e 2jk rl e 2αL e =1 29

30 Lasers: Something for Nothing (almost) at resonance e 2jkL =1 Et t 1 t 2e jkl t 1 t 2 e jkrl e αl = = E i 1 r 1r e 2jkL 2 1 r 1r 2e 2jk rl e 2αL singularity at 1=r 1 r 2 e ΓgL e αil 1=R1R 2 e 2ΓgL 2α i L e E t E i Image is in the public domain 30

31 MIT OpenCourseWare Electromagnetic Energy: From Motors to Lasers Spring 2011 For information about citing these materials or our Terms of Use, visit: