P. Piot, PHYS 630 – Fall 2008

Absorption

• A medium with absorption can be characterized by a complexsusceptibility

• Which implies a complex permittivity

• For a monochromatic wave we have

which we write as

• Considering a wave

the intensity is

Absorption (or attenuation) coefficient

P. Piot, PHYS 630 – Fall 2008

Weakly absorbing medium

• taking refractive index to be related to β via

we have

• Case of weakly absorbing media

expand

• So that

• And finally

/

/

/

/

P. Piot, PHYS 630 – Fall 2008

Strongly absorbing medium

• Case when

• So

• Remembering that

• We get

• Only “+” solution is selected to insure α>0

P. Piot, PHYS 630 – Fall 2008

Resonant medium: Lorentz model

• Consider the differential equation

• Take and then

• Take we get

P. Piot, PHYS 630 – Fall 2008

Resonant medium II

• Real and imaginary part of the susceptibility

• Limiting cases

For

!

"0

#0

$

P. Piot, PHYS 630 – Fall 2008

Resonant medium III

-χ’’χ’

ΔωΔω

χ0

χ0Q

Q=ω0/Δω

P. Piot, PHYS 630 – Fall 2008

Resonant medium IV

• Around the resonance

• For media with multiple resonance the index of refraction is written

(Sellmeier equation)

=Δω

P. Piot, PHYS 630 – Fall 2008

Dispersion

• Dispersion arises from

• Example Snell-Descartes’ law

P. Piot, PHYS 630 – Fall 2008

Phase and Group Velocity

• Group velocity is the velocity of the “envelope” A(t) of a signal. Takethe Fourier transform

• Expand the wavevector

• Gives

• So

the envelope propagates at velocity v

P. Piot, PHYS 630 – Fall 2008

Group Velocity Dispersion (GVD)

• GVD is defined as

• Introducing the group index

• The GVD takes the form

• Or alternatively

P. Piot, PHYS 630 – Fall 2008

Higher Order Dispersion

• Generally can Taylor-expand the wave vector as

• Group velocity

• Group velocity (or delay) dispersion (GVD or GDD)

• Third order dispersion (TOD)

Very important to generate veryshort (

P. Piot, PHYS 630 – Fall 2008

Dispersion effects

• Two cases– Dω>0: normal dispersion– Dω

P. Piot, PHYS 630 – Fall 2008

Dispersion application: chirp pulse amplification

• Produce a pulse• Propagate it in, e.g., a normal dispersion

medium• Amplify it• Compress it using, e.g., an anomalous

dispersion medium

t

ω

amplifier

Chirp mirror

P. Piot, PHYS 630 – Fall 2008

Meta-material: an old idea

• Topic of Friday Colloquium on Nov 11, 2008 (talk by S. Antipov fromANL)

• “Doubly-negative medium”: both the electric permittivity and themagnetic permeability are negative

P. Piot, PHYS 630 – Fall 2008

Meta-material: basics & applications

• In standard materials

• In left-hander materials

• Poynting vector has oppositedirection compared to k vectorconsequence: n

P. Piot, PHYS 630 – Fall 2008

Meta-material: experimental verification I

• Experimental evidence in the microwave regime (~10 GHz)

P. Piot, PHYS 630 – Fall 2008

Meta-material: experimental verification I cnt’d

P. Piot, PHYS 630 – Fall 2008

Meta-material: experimental verification II

• Experimental evidencein the optical regimereported last year…