Negative Propagating Light Daniel Bauer University of Arizona

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1 Negative Prpagating Light Daniel Bauer University f Arizna dbauer@ .arizna.edu Abstract New materials and techniques have presently enabled scientists t significantly alter the speed f light. Recently light has been recrded traveling faster than the speed f light, at s called superluminal velcities. It has als been drastically slwed dwn t a fractin f its riginal speed. Under extreme circumstances, light can als be seen t prpagate backward as time elapses. Rbert Byd is ne f the leading researchers f backward prpagating light. Optics Overview There are at least tw velcities assciated with the prpagatin f light, the grup velcity and the phase velcity. Imagine waves emanating utward frm a rck thrwn in a lake. The cllectin f waves as a whle represents the grup velcity (v g ). The peaks f individual waves represent the phase velcity. Each wave appears t be mving faster than grup. The grup velcity is very imprtant because it defines the speed with which energy r infrmatin is prpagated 1. Under average circumstances bth the grup and phase velcities f light will be psitive and prpagate in the frward directin. Hwever, new materials and techniques have made it pssible t have the phase and grup velcities travel in ppsite directins. The latest science is nw able t increase the grup velcity t greater than the speed f light (v g >c) and t make light travel backwards. Backward Prpagating Light Light behaves as an electrmagnetic wave that scillates as it prpagates frward. The wave mtin f light implies that it has a frequency and wavelength assciated with it that cntributes t its speed. There are tw types f frequency, linear (f) and angular (ω). Linear and angular frequency vary by a cnstant factr f 2 π such thatω = 2πf. In a vacuum, the speed f light (c) is equal t the frequency (f) multiplied by the wavelength

2 c (λ) s that c = fλ. The phase velcity ( v ) equals, where n is the frequency dependant n refractive index. Therefre distinct frequencies will be refracted differently accrding t Snell s Law 2 n 1 sin(θ 1 ) = n 2 sin(θ 2 ). This phenmenn is knwn as dispersin. As illustrated by Graph 1, nrmal dispersin ccurs when the refractive index f the material decreases as the wavelength increases. Anmalus dispersin ccurs when the refractive index increases with wavelength r mathematically when > 0. There is abnrmally dλ high absrptin in regins f anmalus dispersin as illustrated by Figure 1. Figure 1: Tp graph: Strength f absrptin versus wavelength. Bttm graph: Refractive index versus wavelength with the regins f anmalus and nrmal dispersin labeled. Anmalus dispersin is a fundamental part f achieving backward prpagating light. Anmalus dispersin can be used t prduce grup and phase speeds that are in different directins 3. The grup velcity 4 is defined by c v g = (1) n + ω d ω Materials that exhibit large anmalus dispersin allw the grup velcity f the light t exceed c and/r becme negative 5. This is because in the anmalus regin is negative and will therefre cntribute a negative value t the velcity f the grup. dω

3 Equatin 1 predicts backward prpagating light fr the denminatr will be negative. When ω d ω ω is greater than n and d ω values larger than n because n ω is less than d ω 1 the denminatr will becme negative and less than ne. This will result in a negative grup velcity at superluminal speeds. Anmalus dispersin happens in areas f rapid spectral variatin with respect t the refractive index. Therefre, negative values f the grup velcity will ccur in these areas. As a result f the Kramers-Krnig relatinships 6 these regins f rapid spectral variatin are als in the vicinity f a narrw absrptin feature r a narrw dip in a gain feature 7. Near the absrptin peaks, the refractive index f a material takes n a cmplex frm: n = n ik (2) In Equatin 2 k represents the extinctin cefficient and n is the real part f the refractive index 8. The imaginary part f the refractive index, Im(n), represents the absrptin strength f the medium. Kramers-Krnig relatins are imprtant because they allw the real and imaginary parts f the refractive index t be evaluated. They als allw the refractive index t be determined by measuring the absrptin f a material. The Kramers-Krnig relatinships fr the case f backward prpagating light are 6 : 2P ω' Imn( ω') Ren ( ω) + 1 = dω' (3) 2 2 π ω' ω 0 2Pω Re n( ω') + 1 Imn ( ω) = dω' (4) 2 2 π ω' ω 0 Prfessr Byd s team decided t create a decline in the gain curve rather than lk at a narrw absrptin area. The gain (g) is related t the refractive index and the grup index thrugh a series f equatins 7. g I(1 + I) g( δ ) = (5) 1+ I ( T1 δ ) + (1 + I) Where g = initial gain cefficient; δ= the difference between the frequency (ω) f the measured gain value and the frequency (ω ) f a strng wave with intensity (I); I = I/I sat ; T 1 = relaxatin time f the ppulatin inversin 7. The ppulatin inversin

4 refers t an ptical fiber with mre electrns in an excited state than the grund state due t externally applied energy. The relaxatin time is then the time fr the electrns t be restred t a nrmal grund state. The refractive index will then vary spectrally by: g ct1 I δ n( δ ) = n 2 2 (6) hst 2ω1 (1 + I ) ( T1δ ) + (1 + I ) Fr n hst = refractive index f the fiber in use. The index f the grup can be simplified by evaluating it at δ = 0 s that grup refractive index is given by 7. n g ct I 1 g = nhst 3 (7) 2 (1 + I ) Using the set-up frm Figure 2, the directin that light is traveling can be determined. The first part f the experiment determines a reference value fr the utput wavefrm f the 1550 nm pulsed laser. The 980 nm laser is used t establish gain in the E.D.O.F. (erbium dped ptical fiber) which amplifies the signal f the system. The length f the E.D.O.F is repeatedly reduced and measurements are taken. The directin f prpagatin f the wave can then be determined by analyzing the utput wavefrm. Figure 2: An illustratin f a pssible setup fr mnitring the prpagatin f light in a fiber. Fr a fiber with a relaxatin time f 10 ms, g = 1/m and I =.003 the grup velcity will equal This value is very clse t what was recrded in the labratry making slightly different assumptins abut the gain, relaxatin time, and intensity. The negative grup index value shws that, theretically, light can prpagate in the negative directin. These results have been duplicated in the labratry 7.

5 Figure 3: The mtin f a wave as is appraches (1); prpagates thrugh (2); and is transmitted (3) by a piece f glass. Figure 3 illustrates an bserved example f backward prpagating light. As the wave appraches a piece f glass a pulse frms at the ppsite end f the glass (1). This new pulse splits int tw different waves that travel in ppsite directins (2). The backward traveling half races tward the incident light and cancels it ut. The frward mving part cntinues prpagating away frm the medium as transmissin (3). The grup velcity f the wave in the piece f glass is negative. The peak f the transmitted pulse exits the medium prir t the peak f the incident light entering. Figure 4: Nrmalized input and utput wavefrm after prpagatin thrugh an erbium-dped fiber. Inspectin f Graph 4 cnfirms the utput wave is advanced in time with respect t the input wave. The utput wave suffers frm slight defrmatin arund its leading edge. This is represented by the negative wavefrm values at its leading edge. Althugh the wave prpagates backwards, the energy flw, represented by the Pynting Vectr, is

6 psitive fr all cmbinatins f phase and grup velcities. The Pynting Vectr pints in the directin f energy flw 9 and it always pints in the frward directin 3. Therefre infrmatin can nly travel in the frward directin with respect t time. Bigraphy Prfessr Byd attended the Massachusetts Institute f Technlgy and in 1969 graduated with a bachelr s degree in Physics. He then attended the University f Califrnia at Berkeley and in 1977 received his Ph.D. in Physics fr wrks invlving the use f nn-linear ptical systems fr infrared detectin fr astrnmy. Later that year, he was hired by The Institute f Optics at the University f Rchester where he has wrked ever since. In 1987 he was appinted t the Prfessr f Optics psitin. He has written tw bks and ver 200 research papers in additin t being awarded five patents. He is a member f the Optical Sciety f America and the American Physical Sciety. Cnclusin Rbert Byd s grundbreaking research in backward mving light is n the leading edge f technlgical discveries. Light mving backwards in time is an utcme that has been predicted mathematically 4 that has recently been bserved in labratry cnditins. The negative grup velcity f light can be calculated using already knwn equatins fr grup velcity. Furthermre, experimentally determined values f the prpagatin f light agree with thse predicted mathematically. Nrmal materials are nt sufficient t prduce such striking results. In rder t achieve such abnrmal effects materials with high gain are being used. Anmalus dispersin allws fr a negative cntributin t the grup velcity. When this value becmes large enugh it can dminate the equatin fr grup velcity and make it negative. Despite the negative grup velcity the flw f infrmatin is always in the frward directin.

7 Wrks Cited 1- Kasap S.O., Optelectrnics and Phnics Principals and Practices (Prentice-Hall Inc, Upper River Saddle, 2001). 2- Smith, Warren J. Mdern Optical Engineering. (McGraw-Hill, San Francisc, 2000). 3- Gunnar Dlling, Christian Enkrich, Martin Wegener, Cstas M. Sukulis, Stefan Linden, Simultaneus Negative Phase and Grup Velcity f Light in a Metamaterial, Science. 312, (2006). 4- A. Schweinsberg, N. N. Lepeshkin, M.S. Bigelw, R.W. Byd, S. Jarab, Observatin f superluminal and slw light prpagatin in erbium-dped ptical fiber, Eurphysics Letters. 73, (2005). 5- Matthew S Bigelw, Nick N Lepeshkin, Heedeuk Shin, Rbert W Byd, Prpagatin f a smth and discntinuus pulses thrugh materials with very large r very small grup velcities, Jurnal f Physics: Cndensed Matter. 18, (2006) 6- Evangels Chaliass, The Kramers-Krnig relatins fr light mving backwards in time, Athens, Greece. Gr Aegale. 365 Thebes Street 7- Gerge M. Gehring, Aarn Schweinsberg, Christpher Barsi, Natalie Kstinski, Rbert W. Byd, Observatin f a Backward Pulse Prpagatin Thrugh a Medium with a Negative Grup Velcity, Science. 312, (2006). 8- Refractive Index, 10 Nv (accessed) 13 Nv < 9- Pynting Vectr. 10 Nv (accessed) 13 Nv < Vectr>