course 2 : • pt –symmetry basics : the transverse caseias.ust.hk/events/201601wp/doc/lecture...

1IAS PT Symmetry Course 2– Benisty 2016

PT-symmetry and Waveguides / (2) The transverse case

Course 2 : • PT –symmetry basics : the transverse case

• Imperfect PT–symmetry [ plasmonics]

• Application example : switching below the exceptional point

From here on :

Credits to : Anatole Lupu and Aloyse Degiron (IEF, Orsay, France)Mondher Besbes and Jean-Paul Hugonin (my lab : LCF, IOGS, Palaiseau)


My inspiration?


• “The observables of a quantum system are defined to be the self-adjoint operators A on H, a fixed complex Hilbert space.”

Φ |Φ Φ | Φ

• “The Hamiltonian H is the operator associated to the total energy”

• In a finite-dimensional space | ≡ Φ }, the Hamiltonian matrix H = is Hermitian : ∗

• This grants diagonal terms are real in any basis.• Thus granting (eigen)energies are real,

using the basis of H eigenstates | ≡ },

“ The spectrum is real”


• Energy-conserving systems

• Open systemsLosses and/or Gain

from an outside reservoir

Real spectrum

Hermiticity

Non-Hermiticity

Realeigen-values

allowed

? Nature of this boundary

Linear systems


Hermitian operator H ♥ Real Eigenvalues

♥ Real Eigenvalues Hermitian operator

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

@

But Hamiltonians with 1D degree of freedom …… are not that simple

Real E

Harmonic osc


P

Non-Hermitian 2x2 Hamiltonian

♥ Real Eigenvalues … as long as r’’ < s

PT-Symmetric

P

T

s

t t

tt

P

…and T

det

² ²

det 0 → ² whose sign is 0 or 0 !

or ′′


P

Analogy :Time evolution in Quantum mechanics of Two-modes systems

Spatial evolution of two guided modes in z-propagation

Non-Hermitian Hamiltonian Coupled modes equations

♥ Real Eigenvalues … as long as r’’ < s ♥ Real Eigenvalues … as long as g <

PT-Symmetric

P

T

Gain [P sym] loss

gain losses


s

t t

Non conservative "atoms" (?!)(*)

zz

Non-conservative Guides

Evanescent coupling

Evolutionaxis(*) Gain bosons

A1

A2

A1 A2


Losses

ω’ orRe(Energy)

ω'’ orIm(Energy)

Changeofbehaviour ineigenvalues …

…butinhigh‐loss regime

Splitmodes“Toomuchdamping”

Losses


Thecaseofbalanced gain/loss

T11

T12coupling

L

1GAIN

LOSS

P P zx

Exceptionalpoint(EP):singularityof

∂(eigenvalues)/∂(gain)

ω’

Gain/Loss

ω'’

Splitmodes

Gain/Loss

0 0.5 1 1.5 2-2

-1

0

1

2

0 0.5 1 1.5 2-2

-1

0

1

2


Thelight’s bet

T11

T12coupling

LOSS

GAIN…Bettingthatlightgiven toother guide…

Oops,losses

..Willcomebackre‐ amplified

???

Bet works ? Below EP,oscillation,realω.

Bet lost ? overall gainorloss ?What areeigenstates

0 0.5 1 1.5 2-2

-1

0

1

2

Gain/Loss


Eigenstates below/above EP

| |

|c1/c2|

Gain/Loss

| |

| |


eigenstatebehaviourvs.« gain‐loss »

Symmetry breaking,pictorially

●Symmetry-breakingof eigenstates

1

2

AB

A

B("winner-takes-all")


Dangerofunbalanced eigenstates with gain/loss

c|1>+|2> |1>‐ c|2>

(c‐δc)|1>+ δc|2>

-δc|1>+ δc|2>

δc/c)

High growth of unbalanced part

Can be misleading (Reciprocity based on exact eigenmodes)

Loss state Gain state


Trajectory ofeigenvalues incomplex plane

Eigenvalues in complex

plane

EP Re

Im

-1 -0.5 0 0.5 1-2

-1

0

1

2

Note(tracking tool forrealwg,hint from J‐PHugonin):

Productofeigenvalues Π=(ω1ω2)is continuous atEP!Sum Σ=(ω1+ω2) continuous aswell

0 0.5 1 1.5 2-1

0

1

2

3

4


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 »(&verystrongκ)

"named"

Observedwithparametricgain/loss• Rüter et al. (Clausthal u. CREOL, Technion)« Observation of parity–time symmetry in optics », Nat. Phys. 6, 192 (2010)

(tomorrow)


(Transmissionmatrix)

dis

tan

ce L

(µ

m)

log10(T11)

-5-4-3-2-1012345

0 100 200 300

0

200

400

600

800

T11T12

1

.

…

mapof T11

gain (cm-1)

dis

tan

ce L







3FLAVOURSOFIMPERFECTPT ‐SYMMETRY

Losses 2

gain

2=

2 =or

2:fixed,variable

//


ixed losses :Survival ofabrupttransition(a)

(b)

matched losses

fixed (metal) losses

log10(T)-5-4-3-2-1012345

gain (cm-1)

dist

ance

(µ

m)

0 100 200 300

0

200

400

600

800abrupt behaviour

0

200

400

600

800

1000

abrupt behaviour

gain (cm-1)

dist

ance

(µ

m)

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


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 ?


Substrate

BCB

Au

~4 μm~20 nm

~1 cm

Wideband mode (VIS-IR range)

Courtesy of A. Degiron


t

t

Co‐directionalcouplingbetweenplasmonicguideanddielectricguide

Degiron et al, New J. Phys. 2009(Duke U)

BCB

SiO2

Air

Two Eigenmodes

Detuning control by BCB thickness


Experimental waveguides

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


___ Simulations▪ ▪ ▪ Experiments

Measurements vs. coupling length

Arbitrary BCB thickness

Optimized BCB thickness

BCBt=6.6 μm

BCBt=5.4 μm


Dephasing of beating is a first sign

0

0.5

1

0 0.5 1

0

0.7

1.4

0 0.5 1

0

25

50

0 0.5 1

Propagation distance

Tra

nsm

issi

on

T11

T12

g=0 g< gcrit g>> gcrit

Gain g

Losses -g

2

1

2

1

Mdz

di

2/

2/

ig

igM

Coupled Mode Theory







Theexacttransfermatrixofthearbitrarilydetunedcase

∆2

δ12

12

∆

Ω δ ² =M(z)00

cos ΩΩsin Ω

Ωsin Ω

Ωsin Ω cos Ω

Ωsin Ω

exp 2

(Kogelnik 1976)

complex detuningIm detuning

detuning & coupling

Transfer matrix


TheexacttransfermatrixofIm‐onlydetuningcase

∆2

δ ∆

Ω ∆ ²=M(z)

00

cos Ω∆Ω

sin Ω Ωsin Ω

Ωsin Ω cos Ω

∆Ω

sin Ωexp

2


detuning & coupling

Transfer matrix


Theexacttransfermatrixof:Im‐only‐detuning&perfect‐PTcase

∆ δ ∆

Ω=M(z)

00

cos Ω sin Ω sin Ω

sin Ω cos Ω sin Ωexp 0

2

complex detuningIm detuning0 if PT-sym !!

detuning & coupling

Transfer matrix


Theexacttransfermatrixofno‐detuningperfectcase

0

0

∆ 0 δ 0

Ω

=M(z)00

cos sin sin cos


Transfer matrix

Simply rotation


cos Ω sin Ω sin Ω

sin Ω cos Ω sin Ωexp 0

2

Switchingrequirements

cos sin sin cos

∆ 0

1

0

0

1

etc.

∆ Ω

Changeofbeatlength cos&sincancellations

1?

0

cross

⊜bar


Switching andallthetransfer matrix

πsin π

Δ

Ω

exp 2 )

1

0

0

1


BarandCrossperfectswitchstatesinideal(gain=loss)PT‐symmetriccoupler(PTSC)

Bar Cross perfect switch Cross Bar perfect switch

Smallest length Switch

Not good Switch

dB dB

T11 T12T11 T12

2

1 1

0 2

1 0

10 0 2

1 0

1 2

1 1

00 0

T11

T12

T11

T12

⊜ ⊜


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)

So“33%shorter”couplersarepossible

(insofarasmeasuredbyphase)

1.00(usualelectro‐opticswitch’s∆ product)


Whataboutless perfect systems?

2 =or

« 0 »always exist« 1 »level is higherorlower than unity

Butif±5dBaretolerableThen large margin exist !

ExtradBinT12


[1] H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, "Implementation of PT symmetric devices usingplasmonics: principle and applications," Optics Express, vol. 19, pp. 18004-18019, 2011.

[2] H. Benisty, C. Yan, A. T. Lupu, and A. Degiron, "Healing Near-PT-Symmetric Structures to Restore Their Characteristic Singularities: Analysis and Examples," IEEE J. Lightwave Technol., vol. 30, pp. 2675-2683, 2012.

[3] A. Lupu, H. Benisty, and A. Degiron, "Switching using PT symmetry in plasmonic systems: positive role of the losses," Opt. Express, vol. 21, pp. 21651-21668, 6 2013.

[4] A. Lupu, H. Benisty, and A. Degiron, "Using optical PT-symmetry for switchingapplications," Photonics and Nanostructures-Fundamentals and Applications, vol. 12, pp. 305-311, 2014.

[5] H. Benisty, A. Lupu, and A. Degiron, "Transverse periodic PT symmetry for modal demultiplexing in optical waveguides," Phys. Rev. A, vol. 91, p. 053825, 2015.

[6] H. Benisty and M. Besbes, "Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition " J. Appl. Phys., vol. 108, pp. 063108 (1-8), 2010.[7] H. Benisty and M. Besbes, "Confinement and optical properties of the plasmonic inverse-rib waveguide," J. Opt. Soc. Am. B, vol. 29, pp. 818-826, March 29 2012.

References on switching and coupled guides/plasmonics


CONTEXT:Gainwithplasmons

● SPASER(Stockman,Oultonwithnanorods,...)

● OpticalAmplifiers with LRSPP(Berini)


Applicationexample :Plasmonicmodulator

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1 1.2

Normalized Material Gain

Po

we

r (d

B)

SU8 WG

SP WG

•No électro-optic modulation in metals

• PT -symmetry offers a plausible alternative to elaborate active devices.

•No électro-optic modulation in metals

• PT -symmetry offers a plausible alternative to elaborate active devices.

Lupu et al., Opt. Express 2013

28


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


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


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 Single WG ? Gain brings just … loss compensation …Then yeah gain !

n~2

n~1.4


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 ?


« Danger,LASER!»

²

1

=1• Pole at

Well-known Threshold for lasing(no amplification limit within a linear saturation-free ansatz)

Gain Loss

waveguide

ω

No mention of « lasing » or « threshold » ??

• Fabry-Perot formula with gain

• But …

??

Actually a sign that PT-symmetryneeds our work to be a melting-pot !


CONCLUSION

New use of gain beyond plain loss compensation

New entry for open systems (new inner products…) thanks to new quantum theory input

OK for Adaptation to plasmonics (constant loss media)

Basic (transverse) PT-symmetry


[1] H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, "Implementation of PT symmetric devices usingplasmonics: principle and applications," Optics Express, vol. 19, pp. 18004-18019, 2011.

[2] H. Benisty, C. Yan, A. T. Lupu, and A. Degiron, "Healing Near-PT-Symmetric Structures to Restore Their Characteristic Singularities: Analysis and Examples," IEEE J. Lightwave Technol., vol. 30, pp. 2675-2683, 2012.

[3] A. Lupu, H. Benisty, and A. Degiron, "Switching using PT symmetry in plasmonic systems: positive role of the losses," Opt. Express, vol. 21, pp. 21651-21668, 6 2013.

[4] A. Lupu, H. Benisty, and A. Degiron, "Using optical PT-symmetry for switchingapplications," Photonics and Nanostructures-Fundamentals and Applications, vol. 12, pp. 305-311, 2014.

[5] H. Benisty, A. Lupu, and A. Degiron, "Transverse periodic PT symmetry for modal demultiplexing in optical waveguides," Phys. Rev. A, vol. 91, p. 053825, 2015.

[6] H. Benisty and M. Besbes, "Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition " J. Appl. Phys., vol. 108, pp. 063108 (1-8), 2010.[7] H. Benisty and M. Besbes, "Confinement and optical properties of the plasmonic inverse-rib waveguide," J. Opt. Soc. Am. B, vol. 29, pp. 818-826, March 29 2012.

References on switching and coupled guides/plasmonics

course 2 : • pt –symmetry basics : the transverse caseias.ust.hk/events/201601wp/doc/lecture...

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