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New Time Reversal Parities and Optimal Control of Dielectrics for Free Energy Manipulation

Scott GlasgowBrigham Young University, Provo Utah 84602 USA

glasgow@math.byu.edu

Chris Verhaaren University of Maryland, Department of Physics

cver@umd.edu

John CorsonBrigham Young University, Department of Physics

jcorson@byu.edu

Frontiers in Optics 2010 /Laser Science XXVIOctober 24-28, Rochester New York

OSA’s 94th Annual Meeting

Funding—many thanks!

Research facilitated by NSF Grants No. DMS-0453421 and DMS-0755422.

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4

Optimal Control of Dielectric Media:

Optimally Slow and Fast LightNoise reductionUltra-high sensitivity interferometryUltra-high speed and low power optical switchingNetwork traffic managementAll things “all-optical”: buffering, synchronization,

memory, signal processing

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Usual Non-Optimal Approaches to Slow Light—Linear Media

gg

c

n

:g

dnn n

d

Approaches are “frequency local”= narrow band—make index as steep as possible at favorite frequency.

6

Approach: Frequency-global/wide-band analysis of , hence of

Time-Frequency Optimal Approach—Linear Media

3

Conservation Law:

( , ) ( , ) 0 ( ) : ( , ) 0,d d

u t t U t u t dt dt dt

x S x x x

field int

Field and Interaction Energy densities:

( , ) ( , ) ( , ),u t u t u t x x x

int ( , ) [ ](

Interacti

, ) :

on Energy

, [ ]( , )

:t

u t W E t E P E d

x x x x

[ ]( , )W E tx [ ]( , ) [ ; ]( , ).P E P E x x

“Orthogonal decomposition of ”! ( )

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int

Mechanism: interaction energy

created in medium optimally by leading edge

returned

( , ) [ ]( , ) : , [ ]( ,

from medium optimally to trailing edg

)

e

,t

u t W E t E P E d

x x x x

Time-Frequency Optimal Approach=Energy Optimal Approach

Slowing Medium

Slowed Pulse

Unaffected Pulse

Optimal/Broadband design of pulse for medium = energy-minimal excitation + energy maximal de-excitation of medium

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Free Energies of Dielectrics: tutorials from viscoelasticity

• M. Fabrizio and J. Golden, “Maximum and minimum free energies for a linear viscoelastic material,” Quart. Appl. Math. 60, 341–381 (2002).

max[ ]( , ) : min [ ]( , ) : min , [ ]( ,

i.e., minimum energy to creat

) ,

e state creat .ed by

t t

t

E E E EU E t W E t E P E d

E

x x x x

min[ ]( , ) : [ ]( , ) min [ ]( , )

max , [ ]

i.e., maximum energy recoverable from state created

( , ) ,

. by

t

t

t t

E

t t t

Et

U E t W E t W E E

E P E E d

E

x x x

x x

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Unified View of Max and Min Free Energies: Time-reversal

S. G., John Corson and Chris Verhaaren “Dispersive dielectrics and time reversal: Free energies, orthogonal spectra, and parity in dissipative media,” Phys. Rev. E 82, 011115 (2010).

min

:

[ ]( , ) : [ ]( , ) min [ ]( , )

: , [ ]

Maximum energy recoverable from state created

,

by

( ) .

t

t t

E

t t t

t

E

U E t W E t W E E

E P E E d

x x x

x x

Consider only special excitation fields such that the de-excitation field

is exact

ly a m

ultiple of

its time-reversal:

( ) ( ), 0.

t t

t t

E E

E t E t

10

,3. eigenfields: any dielectric staComplete te .tt Span E

Unified View of Max and Min Free Energies: Time-reversal

fact

min max recoverable energy from

EIGENFIELDS: excitation field's de-excitation field is time-reversed

[ ]( , ) : max , [ ]( , ) .

( ) ( ),

multip e

0.

l :

t

t t t t

Et

t t

U E t E E P E E d

E t E t

x x x

1. Rational, passive : only discrete - 1,1 arisetime reversal eigenvalues

2 2, , , ,2. eigenfields: Orthogonal t t t tW aE bE a W E b W E

min max4. generates , . , generat s e .t tE U by definitio By thn E Ueorem

min max , diagonal quadratic for5. and are i .ms n tU U E

min max , 1 , 16. and identical in and ,odd and even fields under

time reversal: and energy.

t t

kinetic potential

U U E E

min max

7. Eigenvalues 1 exist only in multi-resonance systems:

. Corollary: disonant-diss = otherwise.ipation U U

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Global/wide-band analysis of :

Max and Min Free Energies: “classical” E.E. and V.E. theorems

( )

1 1

1 12 2

1 1

rencies

Im ( )( )

: ; : ;

Onsager Causality: Im , 0,

TranRe

spsonances a

N N

j j j

k k k k k k

jj j

N Np p

j

j

k

j

k

k

Z Z

z z z z z z

Z

Z

z

Z Z Z

Symmetry: Im 0 jZ

min min virtual

1virtual 12

1

: max recoverable energy ; , where

( ): ; Onsager Causality: I

m 0

, .

N

jj j

j k

kk

N

kp z z

Z

U U

Z

E

Z z

max max virtual

1virtual 12

1

: min energy to create state ; , where

( ): ; Onsager

k k

j

N

Np

k

jj

Z

U E

Z

z

U

z

Causality: Im , 0.j kZ z

Physical Hypotheses:

Ensuing 2 Theorems:

FUTURE

PAST

12

Global/wide-band analysis of :

Max and Min Free Energies: “classical” E.E. and V.E. theorems

( )

1 1

1 12 2

1 1

rencies

Im ( )( )

: ; : ;

Onsager Causality: Im , 0,

TranRe

spsonances a

N N

j j j

k k k k k k

jj j

N Np p

j

j

k

j

k

k

Z Z

z z z z z z

Z

Z

z

Z Z Z

Symmetry: Im 0 jZ

2min min virtual virtual2

virtual virtual

( ) ; ( ) ; ( ) ,

ˆ ˆwhere ; ( ) ( ) ( )

t

p t

U E t U E t P E d

P E E

2max max virtual virtual2

virtual virtual

( ) ; ( ) ; ( ) ,

ˆ ˆwhere ; ( ) ( ) ( )

t

p t

U E t U E t P E d

P E E

Physical Hypotheses:

Ensuing 2 Theorems:

FUTURE

PAST

13

Global/wide-band analysis of :

Max and Min Free Energies: “classical” E.E. and V.E. theorems

( )

1 1

1 12 2

1 1

rencies

Im ( )( )

: ; : ;

Onsager Causality: Im , 0,

TranRe

spsonances a

N N

j j j

k k k k k k

jj j

N Np p

j

j

k

j

k

k

Z Z

z z z z z z

Z

Z

z

Z Z Z

Symmetry: Im 0 jZ

2irrec virtual min virtual2

virtual virtual

( ) : ; ( ) ( ) ; ( ) ,

ˆ ˆwhere ; ( ) ( ) ( )

t

p

U E t W E t U E t P E d

P E E

2waste virtual max virtual2

virtual virtual

( ) : ; ( ) ( ) ; ( ) ,

ˆ ˆwhere ; ( ) ( ) ( )

t

p

U E t W E t U E t P E d

P E E

Physical Hypotheses:

Ensuing 2 Theorems:

FUTURE

PAST

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Max and Min Free Energies: “classical” E.E. and V.E. theorems

2 2irrec virtual waste virtual2 2

( ) ; ( ) ( ) ; ( )t t

p p

U E t P E d U E t P E d

2 Notions of loss:

3rd Theorem:

FUTURE PAST

"PRESENT"

int ( , ) [ ]( , ) : , [ ]( , )t

u t W E t E P E d

x x x x

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Fast/Slow Light Mixture: Analysis by Max and Min Free Energies

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Time-Reversal and the Effective Susceptibilities: Simplest Examples

S. G., John Corson and Chris Verhaaren “Dispersive dielectrics and time reversal: Free energies, orthogonal spectra, and parity in dissipative media,” Phys. Rev. E 82, 011115 (2010).

2 22 2 2 2 2 2 2 21 1 1 1 1 1

2

1 1

2

Im ( )( )

: : ;

Onsager Causality: Im , , 0,

Resonances4

Transp

renc e

i sap pi

i

i

i

i

i

i

i

1 1

/ 1

Symmetry: Im 0

where

/,; ;F

i

2irrec min virtual2

virtua1 12 2

1 1

l2

: ; ( ) , where

( ): ; Onsager Causality: Im , , 0

t

p

p

i i

U W

i

i

U

i

P E

i

d

2waste max virtual2

vir2

tual

21 1

2

: ; ( ) ,

where

( ): ; Onsager

t

p

p

i

i

U d

i

U W P E

1 1 Causality: Im , , 0ii i

FUTURE

PAST

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Time-Reversal and the Effective Susceptibilities: Simple Example

S. G., John Corson and Chris Verhaaren “Dispersive dielectrics and time reversal: Free energies, orthogonal spectra, and parity in dissipative media,” Phys. Rev. E 82, 011115 (2010).

2 22 2 2 22

2 22

2 2

5Resonances

2 1 10 2 10 1

Im ( )( ) 2.32308

: :

5

rencies4 1 10

Transpa

p pi

i

i

i i

passivity

0

2

2irrec min virtual2

passivityvirtualvirtual

22 2 2 2 2 2 2 22

: ; ( ) , where

323 323Im ( )

2 1 10

5

2 10 1 4 1 1

(:

0

);

t

p

p

U

i

W U E

i

P d

i

0

2waste max virtual2

passivityvirtualvirtual

2 22 2 2 2 2 2 2 22

: ; ( ) ,

whe

2 1 10 2 10 1 4

re

727

5

7279 9Im ( )( )

1:

0;

1

t

p

p

U W U P

i

i

E

i

d

0

FUTURE

PAST

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Time-Reversal and the Effective Susceptibilities: Simple Example

S. G., John Corson and Chris Verhaaren “Dispersive dielectrics and time reversal: Free energies, orthogonal spectra, and parity in dissipative media,” Phys. Rev. E 82, 011115 (2010).

2 22 2 2 22

2 22

2 2

5Resonances

2 1 10 2 10 1

Im ( )( ) 2.32308

: :

5

rencies4 1 10

Transpa

p pi

i

i

i i

passivity

0

FUTURE

PAST

2irrec virtual2

; ( )t

p

U P E d

virtualIm ( )

Im ( )

virtualIm ( )

2waste virtual2

; ( )t

p

U P E d

The creation energy effective susceptibility is always passive for DC, active near positive resonance.

The recoverable energy effective susceptibility is here passive for DC, active “near infinity” . This may be reversed, or it may be passive for all frequencies.

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Time reversal eigenvalues and their susceptibilities forthe example :

Time-Reversal and “Eigen-Susceptibilities”: The Fundamental Theorem

S. G., John Corson and Chris Verhaaren “Dispersive dielectrics and time reversal: Free energies, orthogonal spectra, and parity in dissipative media,” Phys. Rev. E 82, 011115 (2010).

1

01

( )1 ( ) "Kinetic Energy" (Generic)

( )1 ( ) "Potential Energy" (Gene

p

p

i

2 2

1 12 2

1

2 2

1

ric)

( ) "Irreversible Energy" (Spec2 1 1

ial)

,1

0

1

i i

i

2( 1) 2( 1)2 2

min 2 21 1

222( 1) 2( 1)

max 2 2 2 21 1

( ) ( ) ; ( ),2 2

; ( )( )( ) ,

2 2

ˆ ; ( ) : ( ) ( ).

j j

jj

j j

N N

j jp p

N N

j jp j p j

U E t P E t P E t

P E tP E tU E t

P E E

The Fundamental time-reversal orthogonality theorem:

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Summary and To Do:Excitation field=time-reversed multiple of

energetically optimal de-excitation field implies…

1. multiple is special—time reversal eigenvalues

2. excitation field is itself energetically optimal

3. excitations are complete in state space4. excitations are orthogonal with respect

to the work function5. two excitations have even and odd

parity, i.e. eigenvalues +1 and -1, corresponding to potential and kinetic energy, and other parities exist for, and only for, multi-resonance systems.

6. energetic orthogonality gives rise to “orthogonal decomposition of ”

• Current eigenvalues are “spatially local”—useful only for “thin media”

• Compute optimal free space pulse to a) impart energy to “thick” medium most efficiently and then b) extract energy from medium most efficiently—spatio-temporal Carnot cycle

• Inverse problem: what resonance structure allows the above to occur for a simple, narrow-band pulse? Conjecture: likely significantly different than EIT resonance/dissipation structure.

( )