2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 1
Functionalguidedphotonicswithparity‐time(PT)symmetryconcepts
H. Benisty1, A. Lupu2 and A. Degiron2
1Lab Charles Fabry, IOGS, Palaiseau, France
Institut d’Electronique Fondamentale, Univ. PSud, Orsay, France
IOGS
IEF
C2N
BELGIUM
LPN
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 2
WorkS inprogress
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 3
(serious)OUTLINEPT-symmetry brief history & basics
Quantum mech., Exceptional Point, Symmetry-breaking
PT-symmetry & plasmonics « I will survive » Electro-optics with gain&loss Early untold PT symmetry –(long-range plasmons) Example of metal & gain combination
- « PIROW » (plasmonic inverse-rib opt wg) & organics
Switching and PT–symmetry (*) Switching configurations below the Exceptional Point
Multimode waveguides management(**)
(*) A. Lupu, H. Benisty and A. Degiron, Opt. Express 21, 21651 (2013)(**) H. Benisty, A. Lupu, and A. Degiron, Phys. Rev. A 91, 053825 (2015)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 4
P
Analogy inoptics
Hermitian operator H ♥ Real Eigenvalues
♥ Real Eigenvalues Hermitian operator
Non-Hermitian Hamiltonien
Coupled modes equations
C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
♥ Real Eigenvalues … as long as r’’ < s ♥ Real Eigenvalues … as long as g <
PT-Symmetric
T ≡ . . )
Gain [P sym] loss
ε"(x) = ‒ ε "(‒x)
gain losses
PT = PT
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 5
PT‐ symmetry :coupledWGs
Standard case
log10(T)-5-4-3-2-1012345
gain (cm-1)
dist
ance
L (µ
m)
0 100 200 300
0200
400
600
800abrupt behaviour
- g2
T11
T12coupling
L
1g1GAIN
LOSSP
Eigenvalues in complex plane
|T11|
Exceptional point(EP):singularity of
∂(eigenvalues)/∂(gain)
EP
Im() is antisymmetricε"(x) = ‒ ε "(‒x)
EP Re
Im
P
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 6
eigenstatebehaviourvs.« gain‐loss »
Symmetrybreaking
●Symmetry-breakingof eigenstates
12
AB
A
B("winner-takes-all")
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 7
PTsymmetryinOptics
• El Ganainy et al. (CREOL), « Theory of coupled optical PT-symmetric structures », Opt. Lett. 32, 2632 (2007)• Klaiman et al. PRL 2008; Guo et al. PRL 2009; … (topic starts to blow up)
• Ctyroky & Nolting 1996
• 2004-2005 : Kulishov/Greenberg/Poladian/Agarwal:Gratings with Δn= Δnrcos(Kz)+ iΔnicos(Kz+φ), «««nonreciprocity»»»
"unnamed"
"named"
Observedwithparametricgain/loss• Rüter et al. (Clausthal u. CREOL, Technion)« Observation of parity–time symmetry in optics », Nat. Phys. 6, 192 (2010)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 8
PT‐ symmetry
PT-symmetry coupled waveguides PT-symmetry Bragg grating waveguide
Transverse PT-symmetry Longitudinal PT-symmetry
ε"(‒ x) = ‒ ε "(x)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 9
PT‐ symmetry :« nonreciprocal R »
Not genuine nonreciprocity (no µ)
∆ε ~ cos(qx)+i sin(qx) = exp(iqx)
forbiddenallowed
-K K
q
q
π/2 shift of Re(ε)and Im(ε) modulations
Transition betweentwo dispersion branches
-K -K-q
K K-qNo -q! ω
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 10
Reciprocity ( @ Lorentz) =
symmetric S-matrix
YES
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 11
OUTLINEPT-symmetry brief history & basics
Quantum mech., Exceptional Point, Symmetry-breaking
PT-symmetry & plasmonics « I will survive » Electro-optics with gain&loss Early untold PT symmetry –(long-range plasmons) Example of metal & gain combination
- « PIROW » (plasmonic inverse-rib opt wg) & organics
Switching and PT–symmetry (*) Switching configurations below the Exceptional Point
Multimode waveguides management(**)
(*) A. Lupu, H. Benisty and A. Degiron, Opt. Express 21, 21651 (2013)(**) H. Benisty, A. Lupu, and A. Degiron, Phys. Rev. A 91, 053825 (2015)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 12
ixed losses :« Iwill survive »(a)
(b)
matched losses
fixed (metal) losses
log10(T)-5-4-3-2-1012345
gain (cm-1)
dist
ance
(µm
)
0 100 200 300
0200
400
600
800abrupt behaviour
0
200
400
600
800
1000
abrupt behaviour
gain (cm-1)
dist
ance
(µm
)
0 100 200 300
|T11|
|T11|
« passive- PT-structures »
« active- PT-structures »
e.g. Guo et al. PRL 2009
‐ g2
g1GAIN
LOSS
‐ g0
+gGAIN
FIXED LOSS
g0 and shouldbe"matched"
H. Benisty et al. Opt. Express, 19, 18004, 2011
EPremains
EP
g1=‐g2
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 13
PT−
+
_
Kogelnik 1970’s
Electro-optic tuningΔRe(n)
Device length L
Commutation loci
PT-sym, 2010’s
Gain / no gain
Fixed losses (Au) ??
Very few tunableΔRe(n) proposalsin plasmonics
We’d better tune ΔIm(n) !
What are the new diagrams ?
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 14
Substrate
BCB
Au
~4 μm~20 nm
~1 cm
Wideband mode (VIS-IR range)
Courtesy of A. Degiron
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 15
t
t
Co‐directionalcouplingbetweenplasmonicguideanddielectricguide
Degiron et al, New J. Phys. 2009(Duke U)
BCB
SiO2
Air
Two Eigenmodes
Detuning control by BCB thickness
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 16
Experimental waveguides
25
A combination of negative and positive lithography steps are used to fabricate plasmonic stripes coupled to SU8 waveguides, embedded in BCB polymer.A combination of negative and positive lithography steps are used to fabricate plasmonic stripes coupled to SU8 waveguides, embedded in BCB polymer.
CL = coupling length
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 17
___ Simulations▪ ▪ ▪ Experiments
Measurements vs. coupling length
Arbitrary BCB thickness
Optimized BCB thickness
BCBt=6.6 μm
BCBt=5.4 μm
26
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 18
Dephasing of beating is a first sign
0
0.5
1
0 0.5 10
0.7
1.4
0 0.5 10
25
50
0 0.5 1
Propagation distance
Tran
smis
sion
Propagation distance Propagation distance
T11
T12
g=0 g<gcrit g>>gcrit
Gain g
Losses -g
2
1
2
1
M
dzdi
2/2/
igig
M
Coupled Mode Theory
27
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 19
Applicationexample :Plasmonicmodulator
-30
-20
-10
0
10
20
30
0 0.2 0.4 0.6 0.8 1 1.2Normalized Material Gain
Pow
er (d
B)
SU8 WG
SP WG
•No electro-optic modulation in metals• PT -symmetry offers a plausible alternative to elaborate active devices.
•No electro-optic modulation in metals• PT -symmetry offers a plausible alternative to elaborate active devices.
Lupu et al., Opt. Express 2013
28
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 20
Moreminiatureplasmonics?Example ofPlasmonic Inverse‐Rib OpticalWG
JAP 2010, H. Benisty and M. BesbesE-field in 30-50 nm tip...(+JOSA 2012)
Like Oulton’s nanorod/spaser, but deterministic Gain brings just gain !
n~1.4n~2.0
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 21
Two coupled «PIROWs »:good?Again oneandanormal one
CanwehaveagoodEP?
H. Benisty and M. Besbes,
"Confinement and optical properties of the plasmonicinverse-rib waveguide,"
JOSA. B, vol. 29, pp. 818-826, 2012.
GAIN FIXED LOSSES
METAL
Something wrong ?
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 22
« Healing »ofTwo Coupled PIROWs
=-0.174
Asmall detuning compensate complex (Re&Im)couplingconstant IEEE J. Lightwave Technol., vol. 30, pp. 2675-2683, 2012. (H. B et al.)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 23
ModelRealisation :Gainwith organics
Thermally evaporated “fvin” layer on glass substrate
Adjustable razor blades
Diverging LensCylindrical Lens
Sample
CCD(stripe lengthmeasurement)
Stripe quality (a) Without imaging (b) With imaging
Pump LASER Characteristics
Frequency Doubled Q-switched Nd:YVO4λ = 532 nm10 Hz, pulse duration < 500 ps
Pump Stripe Width : 320 µm
Variable Stripe Length (VSL) technique:+ Molecular film
Spectro(Ocean Optics)Spectro(Ocean Optics)Spectro(Ocean Optics)
(b)
(a)
“fvin”
H. Rabbani-Haghighi, S. Forget, S. Chénais et al. Appl. Phys. Lett. 95, 033305 (2009).
S.Chénais &S.Forgetteam,LPL,Paris13
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 24
E=0,41 mJ/cm2 → Gain= 50 cm‐1
Comparison : in a 5%‐doped DCM:PMMA of same thickness :
Gain= 40 cm‐1
Gainwith organics can be high&fast
nanosec. pump‐probe gain
S.Chénais &S.Forgetteam,LPL,Paris13
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 25
OUTLINEPT-symmetry brief history & basics
Quantum mech., Exceptional Point, Symmetry-breaking
PT-symmetry & plasmonics « I will survive » Electro-optics with gain&loss Early untold PT symmetry –(long-range plasmons) Example of metal & gain combination
- « PIROW » (plasmonic inverse-rib opt wg) & organics
Switching and PT–symmetry (*) Switching configurations below the Exceptional Point
Multimode waveguides management(**)
(*) A. Lupu, H. Benisty and A. Degiron, Opt. Express 21, 21651 (2013)(**) H. Benisty, A. Lupu, and A. Degiron, Phys. Rev. A 91, 053825 (2015)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 26
BarandCrossperfectswitchstatesinideal(gain=loss)PT‐symmetriccoupler(PTSC)
Bar Cross perfect switch Cross Bar perfect switch
Smallest length SwitchNot good Switch
dB dB
T11 T12T11 T12
g1
2
1 1
0
g1
2
1 0
10 0
g1
2
1 0
1
g1
2
1 1
00 0
T11
T12
T11
T12
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 27
2
2 2 2im 2 2
im
tan 1LL
im 0.67 2.103L
CMTsolution
Whatisthenewconceptreplacing“Vπ”?
Realindexmodulation Imaginaryindexmodulation
Δ(Im)replacesΔ(Re)
Δ(Im)=|g|+|χ|(the« sum »ofgainandloss)
Sorathershortcouplersarepossible
[#¤@§§](*)
(*)Copyrighted
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 28
FLAVOURSOFIMPERFECTPT ‐SYMMETRY
Losses2
gaing1
2=g1
2 =g1or
2:fixed,variableg1
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 29
1exp 2 cg L
Positive role of losses : less gain needed Unbalanced coupler penalty
29
1 2im 2 0.67
2 cgL L
Remarkable feature
Switching possible with less gain and more losses, provided @0.67π met
UnbalancedPTSC:positiveroleoflosses!
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 30
OpticalMemoryconfiguration
Kulishov 2005
Concept remains to be proven…
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 31
OUTLINEPT-symmetry brief history & basics
Quantum mech., Exceptional Point, Symmetry-breaking
PT-symmetry & plasmonics « I will survive » Electro-optics with gain&loss Early untold PT symmetry –(long-range plasmons) Example of metal & gain combination
- « PIROW » (plasmonic inverse-rib opt wg) & organics
Switching and PT–symmetry (*) Switching configurations below the Exceptional Point
Multimode waveguides management(**)
(*) A. Lupu, H. Benisty and A. Degiron, Opt. Express 21, 21651 (2013)(**) H. Benisty, A. Lupu, and A. Degiron, Phys. Rev. A 91, 053825 (2015)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 32
Multimodetransmission
without crosstalk ! short datacom
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 33
m=1,2,3,... How to select one mode (at given ω)?
ByGrating (+Lens)?ByMMIvariant?Bygrating variants ?
Castro et al.Liu, Miller and Fan, S.H. Cheng, (2012) et al.
Modeselection recipes ?
?Confined hologram ?
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 34
Three flavours ofPT‐symmetry
Periodicity(bounded)
& PT-symmetry
propagation
propagation
PT-symmetryPeriodicitypropagation& PT-symmetry
Kulishov« pseudo -nonreciprocal… »
Simple case
New case
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 35
-0.02-0.01
00.01
0.020.03
0.040.05
0.06
1000
2000
3000
4000
5000
6000
2.80
2.70
2.60
0 0.1 0.2 0.3 0.4 0.5
| ΔεI |
Re(n
eff)
zeros
#1
#6 #7
0.1 0.2 0.3 0.4 0.50-0.02
0
0.02
0.04
0.06
| ΔεI |Im
(nef
f)
Modesby1DTMMcalculation
ΔεI
Phys. Rev. A 91, 053825 (2015)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 36
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
10-4
10-2.2
10-3
eige
nvec
tori
ndex
1
31
61
1 31 61
1 12 24 36
|<ψ | ΔεI |ψ >|²
2nd order perturbationtheory explains modeclusters
Phys. Rev. A 91, 053825 (2015)
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 37
9 stripes gain/loss pattern
0 4 8
1.2
1.199
1.198
|ΔωI| 104
Im(ω
)10
4
(a) 5
0
-50 4 8
|ΔωI| 104
(b)
Re(ω
)loss mode
gain mode
gain&loss
modes
0 72 144
-3.14
0
3.14
0
1
2
Transverse coordinate x
Ampli
tude
(a.u.
)Ph
ase
loss modegain mode
(d)
(e)
0 5 10 15n°4
0 5 10 150
0.51
-101
n°5
0 5 10 150
0.51
-101
n°6
Only modes#4and#5gotobroken sym
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 38
Fieldevolution with PT‐sym/just‐sym stripes
……
PT‐sym P‐only‐sym4 gainlobes
P‐only‐sym4 loss lobes
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 39
Work not(notyet)inprogress
2015 IEEE Benelux Chapter,May 22nd, Mons, H. Benisty et al. 40
CONCLUSION
PT–symmetry
« Hologram » of gain and losses
Many Emerging features @ Exceptional Point.
Several avenues for worthwile photonic devices
Plasmonics Replace Electro-optics by Gain&LossGood switching conditions existbelow Sym.-Breaking point.
Transverse mode management