Optics

Speed of Light and Angle of Propagation in an Absorbing Medium : 

1 Speed of Light and Angle of Propagation in an Absorbing Medium William H. Southwell Table Mountain Optics 509 Marin Street Suite 125 Thousand Oaks, California 91360 bill@tablemountainoptics.com OIC TuA8 9:15 pm 8-June-2010 Tucson, Arizona

Abstract : 

Abstract We show that light travels faster than c as it traverses thin layers of silver and gold and other materials. Also presented is an expression for the real angle of refraction in absorbing media. 8-June-2010 Table Mountain Optics 2

What is n ? Refractive Index : 

What is n ? Refractive Index 8-June-2010 Table Mountain Optics 3 It is: n = c/v Where v = speed of light in the material Where c = speed of light in vacuum But some materials have n < 1 Does that mean v > c ? The speed of light faster than in a vacuum? But, these materials also have k  0 Is there some interaction between n and k that changes the speed of light? Does it prevent v > c? Yes Yes, with angle No

Published n and k for Silver (Palik) : 

Published n and k for Silver (Palik) 8-June-2010 Table Mountain Optics 4

Maxwell’s Equations : 

Maxwell’s Equations 8-June-2010 Table Mountain Optics 5 ∆×H=j+∂D/∂t ∆×E=-∂B/∂t ∆∙D=ρ ∆∙B=0 To these we add the material equations: j=σE D=εE B=μH Performing the indicated partial derivatives and lumping constants K2=εμω2-i μσω . N2=εμc2-iμσc2/ω Obtained is a wave equation with solution E=E exp[i(ωt-K∙r) ]

Damped Plane Wave : 

Damped Plane Wave 8-June-2010 Table Mountain Optics 6 The lumped constant is complex N = n – ik Which has the following meaning: E=E exp(-2kz/λ )exp[i(ωt-2ns/λ) ] Following a point of constant phase, n is related to the phase velocity v of the wave v = c/n phase velocity of the wave k extinction coefficient

Apply Boundary Conditions to Find the Wave in the Absorbing Medium : 

Apply Boundary Conditions to Find the Wave in the Absorbing Medium 8-June-2010 Table Mountain Optics 7 E0 = E0 exp[i{ωt - (2n0)/λ (x sin ϑ0 + z cos⁡ ϑ0 )}] Er = Er exp[i{ωt - (2n0)/λ (xr + yβr + zr )}] E = E exp[i{ωt - 2(n’ – ik’)/λ (x+yβ+z)}]. E = E exp(-2ktz/λ )exp[i{ωt-(2nt/λ) (x sin⁡(ϑt) + z cos⁡(ϑt))}]

The Solution : 

The Solution 8-June-2010 Table Mountain Optics 8 E = E exp(-2ktz/λ )exp[i{ωt-(2nt/λ)(x sin⁡(ϑt) + z cos⁡(ϑt)) }] nt2 = ½[√{4n’2k’2 + (n’2 - k’2 - n02sin2ϑ0)2} + n02sin2ϑ0 + (n’2 - k’2)] kt2 = ½[√{4n’2k’2 + (n’2 - k’2 - n02sin2ϑ0)2} + n02sin2ϑ0 - (n’2 - k’2)] nt sinϑt = n0 sinϑ0 ϑt is the real angle of propagation nt is the real effective index v = c/nt phase velocity of the wave kt extinction coefficient (normal to surface)

Interesting Properties : 

Interesting Properties nt2 - kt2 = n’2 - k’2 2nt kt = no easy expression ηs = (nt2 - n02sin2ϑ0)1/2 – i kt Generalized Tilted Admittance ηp = no easy expression 8-June-2010 Table Mountain Optics 9

Wave Propagating in an Absorbing Medium : 

Wave Propagating in an Absorbing Medium Damped propagating wave Damping is along z , normal to the surface Damping is according to “effective” kt Wave propagates with “effective” nt Wave propagates along direction s s= x sin⁡(ϑt)i + z cos⁡(ϑt))j 8-June-2010 Table Mountain Optics 10 E = E exp(-2ktz/λ )exp[i{ωt-(2nt/λ)(x sin⁡(ϑt) + z cos⁡(ϑt)) }]

Generalization of Snell’s Law : 

Generalization of Snell’s Law 8-June-2010 Table Mountain Optics 11 Applies to absorbing media (n’ , k’ ) normal incidence Transmitted beam angle ϑt is real Effective index nt is real but depends on n’ , k’ , and ϑ0 The beam in an absorbing media remembers how it got in: Its speed depends on ϑ0 nt sin⁡(ϑt) = n0 sin⁡(ϑ0)

Propagation Angle θt in Ag : 

Propagation Angle θt in Ag 8-June-2010 Table Mountain Optics 12

Single Surface Deflection Angle θt – θ0 : 

Single Surface Deflection Angle θt – θ0 8-June-2010 Table Mountain Optics 13

Limiting Cases : 

Limiting Cases At normal incidence ϑ0 = 0, nt = n’ and kt = k’ , which says that the speed of light at normal incidence is determined by the real part of the medium refractive index, that is, the light speed at normal incidence is independent of k’. As k’ goes to zero nt = n’ and kt = 0 for any angle of incidence, which is the case for usual refraction and propagation in dielectric materials. 8-June-2010 Table Mountain Optics 14

Effective n and k for Silver : 

Effective n and k for Silver 8-June-2010 Table Mountain Optics 15

Speed of Light in Silver at 550 nm : 

Speed of Light in Silver at 550 nm 8-June-2010 Table Mountain Optics 16

Group Velocity is Even Faster : 

Group Velocity is Even Faster 8-June-2010 Table Mountain Optics 17 dn/dλ is positive for Ag mid-Visible to NIR Anomalous dispersion Thus vg > v > c

Ag is Essentially Non-Dispersive Visible : 

Ag is Essentially Non-Dispersive Visible 8-June-2010 Table Mountain Optics 18

Phase and Group Velocity in Silver : 

Phase and Group Velocity in Silver 8-June-2010 Table Mountain Optics 19

Results : 

Results Light in an absorbing medium is attenuated But it does not slow down Thus v = c/n is precisely correct Light travels faster than c in gold and silver and other nitrides and oxides 8-June-2010 Table Mountain Optics 20

Huygens’ wavelets for n>1 (left) and n<1 (right) : 

8-June-2010 Table Mountain Optics 21 Huygens’ wavelets for n>1 (left) and n<1 (right)

Speed of Light Determined From Prism Deflection : 

Speed of Light Determined From Prism Deflection 8-June-2010 Table Mountain Optics 22 An equation for effective index nt in the prism from the measurement of the incidence angle, the deflection angle δ, and the prism apex angle ϵ: nt= √{sin2 ϑ0+[(cos⁡ ϵ sinϑ0 + sin⁡(ϵ-ϑ0-δ) )/sin ⁡ϵ ]2} If the incident angle is chosen to be zero, this reduces to: nt= sin⁡(ϵ-δ)/sin⁡ϵ For a thin film with a small wedge angle ϵ the speed of light is determined from: v=c/(1 – δ/ϵ)

Prism or thin film wedge : 

Prism or thin film wedge 8-June-2010 Table Mountain Optics 23 v=c/(1 - δ/ϵ)

Experimental Verification in 1888 : 

Experimental Verification in 1888 Experiments on thin silver wedges were done by A. Kundt in 1888. Results are given for silver, copper, platinum, iron, nickel, and bismuth. Silver, gold, and copper were found to have negative deflections (toward the thin side of the wedge). Based on the measured deflection angle he obtained an index of refraction for Ag of 0.27. He could not detect a significant difference for various colors indicating low dispersion, which is in agreement with current tables. Here is what he says, “The velocity of light in silver is nearly four times as great as in vacuo, but the dispersion in silver is not very great.” Kundt, A. “On the indices of refraction of the metals,” The London, Edinburch, and Dublin Philosophical Magazine and Journal of Science 5th Series, 1-18 (1888). Translated from Sitxungsberichte der kön. Preuss. Akad. Der Wissenschaften, Feb. 16, 1888. See also Kundt, A. Ann. d. Physic, Vol. 34, 469 (1888). 8-June-2010 Table Mountain Optics 24

But what about the speed of information? : 

But what about the speed of information? It has been argued that an infinite monochromatic plane wave must be perturbed in some way and this perturbation must propagate in order to convey information. But any such perturbation, it is claimed, destroys the infinite harmonic wave, making phase velocity impossible to measure and void of significance. Born, M. & Wolf, E. Principles of Optics, 4th edn, Sec. 1.3.3 (Pergamon Press, New York, 1970). It is apparently a strongly held belief in modern literature that information cannot travel faster than c. 8-June-2010 Table Mountain Optics 25

Suggested Experiment : 

Suggested Experiment Construct wedged thin Ag film Establish that light deflects toward apex Modulate the beam Intensity Frequency Pulses Observe if deflection is away from apex during modulation If not, then information travels faster than c 8-June-2010 Table Mountain Optics 26

Measure Beam Displacement of Thin Film : 

Measure Beam Displacement of Thin Film 8-June-2010 Table Mountain Optics 27 The sign of the offset determines: v > c

Beam Offset Δ for a Multilayer with Absorbing Layers (Prompt Ray Only) : 

Beam Offset Δ for a Multilayer with Absorbing Layers (Prompt Ray Only) 8-June-2010 Table Mountain Optics 28 Prompt ray is Snell’s Law ray First one through May not be the center of the beam due to multiple reflections

Prompt Ray Offset for MDMDM Induced Transmission Filter (120 nm Ag, 275 nm SiO2) : 

Prompt Ray Offset for MDMDM Induced Transmission Filter (120 nm Ag, 275 nm SiO2) 8-June-2010 Table Mountain Optics 29

Further Work : 

Further Work Separate s- and p- components at angle Two interfaces and multiple reflections at an angle Will the characteristic matrix require the use of nt , angle dependence? Can ellipsometry be used to measure the angle dependence of nt? 8-June-2010 Table Mountain Optics 30

Summary : 

Summary 8-June-2010 Table Mountain Optics 31 n and k are the fundamental optical properties of materials n is a measure of the speed of light Light travels faster than c when n < 1 Silver, gold, some nitrides and oxides Thin prism experiments will establish that phase velocity, group velocity, information velocity all travel > c