New Time Reversal Parities and Optimal Control of
Dielectrics for Free Energy Manipulation
Scott Glasgow
Brigham Young University, Provo Utah 84602 USA
glasgow@math.byu.edu
Chris Verhaaren
University of Maryland, Department of Physics
cver@umd.edu
John Corson
Brigham Young University, Department of Physics
jcorson@byu.edu
1
Frontiers in Optics 2010 /Laser Science XXVI
October 24-28, Rochester New York
OSA’s 94th Annual Meeting
Funding—many thanks!
Research facilitated by NSF Grants No. DMS0453421 and DMS-0755422.
3
Optimal Control of Dielectric Media:
Optimally Slow and Fast Light
Noise reduction
Ultra-high sensitivity interferometry
Ultra-high speed and low power optical switching
Network traffic management
All things “all-optical”: buffering, synchronization,
memory, signal processing
4
Usual Non-Optimal Approaches to Slow
Light—Linear Media
c
g 
ng
dn
ng : n  
d
Approaches are “frequency local”= narrow
band—make index as steep as possible at
favorite frequency.
5
Time-Frequency Optimal Approach—Linear
Media
Conservation Law:

d
d
u (x, t )    S(x, t )  0  U (t ) :  u (x, t )d 3x  0,
t
dt
dt
Field and Interaction Energy densities:
u (x, t )  ufield (x, t )  uint (x, t ),
Interaction Energy:
t
uint (x, t )  W [ E ](x, t ) :
 E  x,  P[ E ](x, )d

Approach: Frequency-global/wide-band
analysis of W [ E ](x, t ) , hence of P[ E](x, )  P[ E;  ](x, ).
“Orthogonal decomposition of    ( ) ”!
6
Time-Frequency Optimal Approach=Energy
Optimal Approach
Slowed Pulse
Unaffected Pulse
Optimal/Broadband design of
Slowing Medium pulse for medium = energyMechanism: interaction energy
minimal excitation + energy
t
maximal de-excitation of
uint (x, t )  W [ E ](x, t ) :  E  x,  P[ E ]( x, )d , medium

created in medium optimally by leading edge
returned from medium optimally to trailing edge
7
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).
t

  
U max [ E ](x, t ) : mint W [ E ]( x, t ) : mint
 
E E
E E
E  x,   P[ E ]( x,  )d ,
i.e., minimum energy to create state created by E.
t
t
U min [ E ]( x, t ) : W [ E ]( x, t )  min
W
[
E

E
]( x, )


t
E
  t

t
t
 max
   E  x,  P[ E  E ](x, )d  ,
Et
t


i.e., maximum energy recoverable from state created by E.
8
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).
Maximum energy recoverable from state created by E :
t
t
U min [ E ](x, t ) : W [ E ](x, t )  min
W
[
E

E

 ]( x, )
t
E

:   Et  x,  P[ Et  Et ](x, )d .
t
Consider only special excitation fields Et such that the de-excitation field Et
is exactly a multiple  of its time-reversal:
Et (t   )   Et (t   ),   0.
9
Unified View of Max and Min Free
Energies: Time-reversal
U min [ E ](x, t ) : max recoverable energy from   E
t


  max   E  x,  P[ E
fact
Et
t

t

 Et ]( x, )d .
t
EIGENFIELDS: excitation field's de-excitation field is time-reversed multiple:
Et (t   )   Et (t   ),   0.

1. Rational, passive  : only discrete time - reversal eigenvalues   1,1 arise
2. Orthogonal eigenfields: W  aEt ,  bEt ,    a 2W  Et ,   b 2W  Et ,  
3. Complete eigenfields: any dielectric state t  Span Et ,  .

t
t
4. E generates U min , by definition. By theorem, E generates U max .
5. U min and U max are diagonal quadratic forms in Et , .
 

t
 , 1
6. U min and U max identical in E
t
 , 1
and E
,odd and even fields under
time reversal: kinetic and potential energy.
7. Eigenvalues   1 exist only in multi-resonance systems:
disonant-dissipation. Corollary: U min = U max otherwise.
10
Max and Min Free Energies: “classical”
E.E. and V.E. theorems
Global/wide-band analysis of
   ( )
:
Physical Hypotheses:
Im    ( )
   Z j    Z j  
N
Resonances:
 ( )

 p2
j 1
N 1
   z    z 
k 1
;
Transparencies:

k
k


j 1
N 1
j
j

j
j
   z    z    z    z 
k 1

k
k

k
k
Symmetry: Im  Z j   0
 
Onsager Causality: Im  Z j , zk   0,
Ensuing 2 Theorems:

  p2
   Z    Z    Z    Z  
N
U min : max recoverable energy  U min  E;  virtual  , where
   Z    Z  
N
 virtual ( )
:
 p2
j 1
N 1

j
j
   z    z 

k
k 1
FUTURE
; Onsager Causality: Im  Z j , zk   0.


k
U max : min energy to create state  U max  E;  virtual  , where
   Z    Z  
N
 virtual ( )
:
 p2
j 1
N 1

j
j
   z    z 
k 1

k
k
PAST
; Onsager Causality: Im  Z j , zk   0.


11
;
Max and Min Free Energies: “classical”
E.E. and V.E. theorems
Global/wide-band analysis of
   ( )
:
Physical Hypotheses:
Im    ( )
   Z j    Z j  
N
Resonances:
 ( )

 p2
j 1
N 1
   z    z 
k 1

k
Onsager Causality: Im  Z j , zk   0,
Ensuing 2 Theorems:
;
Transparencies:

  p2
k
   Z    Z    Z    Z  
N


j 1
N 1
j
j

j
j
   z    z    z    z 
k 1

k
k

k
k
Symmetry: Im  Z j   0
 
  2 t
U min  E  (t )  U min  E;  virtual  (t )  2  P  E ;  virtual  ( )d ,
p t
where Pˆ  E;  virtual  ( )   virtual ( ) Eˆ ( )
FUTURE
  2 t
U max  E  (t )  U max  E;  virtual  (t )  2  P  E ;  virtual  ( )d ,
p t
where Pˆ  E;  virtual  ( )   virtual ( ) Eˆ ( )
PAST
12
;
Max and Min Free Energies: “classical”
E.E. and V.E. theorems
Global/wide-band analysis of
   ( )
:
Physical Hypotheses:
Im    ( )
   Z j    Z j  
N
Resonances:
 ( )

 p2
j 1
N 1
   z    z 
k 1
;
Transparencies:

k
Onsager Causality: Im  Z j , zk   0,
   Z    Z    Z    Z  
N

  p2


j
j 1
N 1
j

j
j
   z    z    z    z 
k

k
k 1
k

k
k
Symmetry: Im  Z j   0
 
Ensuing 2 Theorems:

U irrec  E  (t ) : W  E;  virtual  (t )  U min  E  (t )  2
p
t
 P  E; 
2
 ( )d ,

where Pˆ  E;  virtual  ( )   virtual ( ) Eˆ ( )

U waste  E  (t ) : W  E;  virtual  (t )  U max  E  (t )  2
p
where Pˆ  E;  virtual  ( )   virtual ( ) Eˆ ( )
virtual
FUTURE
t
 P  E; 
2
virtual
 ( )d ,

PAST
13
;
Max and Min Free Energies: “classical”
E.E. and V.E. theorems
2 Notions of loss:
3rd Theorem:

U irrec  E  (t )  2
p
FUTURE
PAST

P
E
;

(

)
d


U
E
(
t
)





virtual
waste

 p2

t
t
2
"PRESENT"
 P  E; 
2
virtual
 ( )d

t
uint (x, t )  W [ E ](x, t ) :
 E  x,  P[ E](x, )d

14
Fast/Slow Light Mixture: Analysis
by Max and Min Free Energies
15
Time-Reversal and the Effective
Susceptibilities: Simplest Examples
Resonances:
Im    ( )
 ( )
  i

2
 p   i    i 1 2  12 


Transparencies:

  p2
  i  i

2
     2  12   12   4 1212 
Symmetry: Im  i  0

Onsager Causality: Im  i, i , 1  i 1   0,
2
2
where
 / 1
  F  ;  1 , 1 ;  /    
U irrec : W  U min
FUTURE

 2
p
t
 P  E; 
2
virtual
 ( )d , where

 virtual ( )
  i 
:
; Onsager Causality: Im  i, i , 1  i 1   0
2
2
2
 p

  i    i 1   1

U waste : W  U max
PAST

 2
p

t
 P  E; 
2
virtual
 ( )d ,

where
 virtual ( )
  i 
:
; Onsager Causality: Im  i, i , 1  i 1   0
2
2
2
 p



i



i




 
1
1


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).
16
;
Time-Reversal and the Effective
Susceptibilities: Simple Example
Resonances:
 ( )
  2.32308i

2
 p   2i    1i 2  102 

Im    ( )
 Transparencies:


  p2

  5i   5i 

2
2
 22   2  102  12   4 12 102 


passivity
 0
FUTURE
U irrec : W  U min

 2
p
t
 P  E; 
2
virtual
 ( )d , where




   323   323
Im   virtual  ( )
 virtual ( )
  5i
:
;

2
2
 p2

  2i    1i   102 
 2  22   2  102  12   4 12 102 

U waste : W  U max
PAST

 2
p

t
 P  E; 
2
virtual

passivity
 0

 ( )d ,

where

727 
727 
  




9 
9 
Im   virtual  ( )
 virtual ( )
  5i

:
;

2
2
2
 p2


  2i    1i   10 
 2  22   2  102  12   4 12 102 
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).
passivity
 0
17
Time-Reversal and the Effective
Susceptibilities: Simple Example
Resonances:
 ( )
  2.32308i

2
 p   2i    1i 2  102 

Im    ( )
 Transparencies:


  p2

  5i   5i 

2
2
 22   2  102  12   4 12 102 


passivity
The creation energy
 0 effective susceptibility
is always passive for DC, active near
positive resonance.
FUTURE
Im   virtual  ( )
Im   virtual  ( )


Im    ( )

PAST
The recoverable energy effective
 t susceptibility is here passive for DC, active
 t
U waste  2  P 2  E;  virtual  ( )d
U irrec  2  P 2  E;  virtual  ( )d
 p  “near infinity” . This may be reversed, or it
 p 
may
be passive
all frequencies.
S. G., John Corson and Chris Verhaaren “Dispersive
dielectrics
and time for
reversal:
Free energies,
orthogonal spectra, and parity in dissipative media,” Phys. Rev. E 82, 011115 (2010).
18
Time-Reversal and “Eigen-Susceptibilities”:
The Fundamental Theorem
The Fundamental
time-reversal
orthogonality
theorem:
Time reversal
eigenvalues and
their
susceptibilities for
the example :

U min  E  (t )  2
2 p
2( N 1)

U max  E  (t )  2
2 p
2( N 1)
P2j  E  (t )
j 1
 j2

2( N 1)
j
j 1


P  E (t )  2
2 p
2

 2
2 p

j 1
P 2  E ;   j  (t ),


2( N 1)

P 2  E ;   j  (t )
j 1
 j2
,
Pˆ  E;   j  ( ) :   j ( ) E ( ).


  1
 1 ( ) 
  1
 1 ( ) 

  ( ) 
i ( )
p
0  ( )
p
i
  2i    1i   102 
2

"Kinetic Energy"
(Generic)
"Potential Energy"
(Generic)
"Irreversible Energy" (Special)

2
       1   1

  1,1
       1 2  12
2
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).
19
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 space
4.
excitations are orthogonal with respect
to the work function
5.
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.
20