H. Benisty A. Lupu2 and A. Degiron 2« Non-hermitian photonics in complex media », Crete, 15-18 June, H. Benisty et al. 1 Multimode waveguide modal management with PT‐ symmetry

« Non-hermitian photonics in complex media », Crete, 15-18 June, H. Benisty et al. 1

MultimodewaveguidemodalmanagementwithPT‐symmetry

H. Benisty1, A. Lupu2 and A. Degiron2

1Lab Charles Fabry, IOGS, Palaiseau, France

Institut d’Electronique Fondamentale, Univ. PSud, Orsay, France


eigenstate behaviour vs.« gain‐loss »

Symmetrybreaking

●Symmetry-breakingof eigenstates

1

2

AB

A

B("winner-takes-all")


OpticalMemoryconfiguration

Kulishov 2005

Concept remains to be proven…


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)


PT‐ symmetry

PT-symmetry coupled waveguides PT-symmetry Bragg grating waveguide

Transverse PT-symmetry Longitudinal PT-symmetry

ε"(‒ x) = ‒ ε "(x)


OUTLINE

Multimode vs. Monomode transmission in waveguides Multimode devices (pre-/post- photonic crystals) Multimode transmissionAgile multimode devices

Transverse PT symmetry of multimode waveguide Eigenmodes and eigenvalues of realistic and effective model Transverse wavevector selection rule Modal selection with the exceptional point

Transverse PT symmetry with static disorder Can one apply « some transverse k » to a WG ? The medium is the message (?) : Transverse k-filters Longitudinal adiabaticity and (lack of) reciprocity Connection with random matrix framework ?

(*) 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)


Multimodetransmission

without crosstalk ! short datacom


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 ?


Multimode‐based frequency demux

(demux ofFdtl modebycoupling toahigher‐order mode)

Coll. FUNFOX (2004-2008)

Coll. Ghent/IMEC (2009)

« PHOTON SIEVE »


Multimodemanagementwith real‐index(2 2MMIconfiguration)


OUTLINE



Transverse PT symmetry with static disorder Can one apply « some transverse k » to a WG ? The medium is the message (?) : Transverse k-filters Longitudinal adiabaticity and (lack of) reciprocity Connection with random matrix framework ?



Three flavours ofPT‐symmetry

Periodicity(bounded)

& PT-symmetry

propagation

propagation

PT-symmetryPeriodicitypropagation& PT-symmetry

Kulishov« pseudo -nonreciprocal… »

Simple case

New case


-0.0

2-0

.01

00

.01

0.0

20

.03

0.0

40

.05

0.0

6

10

00

20

00

30

00

40

00

50

00

60

00

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)


0 1 2

(b) (c)

0 4 81000 x perturbation

1.20

1.18

1.16

1.200

1.196

1000 x perturbation

Eig

enva

lueω

|<ψ | ΔεI |ψ >|²


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

tor

inde

x1

31

61

1 31 61

1 12 24 36

|<ψ | ΔεI |ψ >|²

2nd order perturbationtheory explains modeclusters

Phys. Rev. A 91, 053825 (2015)


0 2 4

1.1992

1.1996

1.2

|ΔωI| 104

Re(ω

)

x

Im(ε)

Fixed losses/Variable gaindistribution

(a) (b)

(c)

Re(ω)

0

-2

1.19921.19961.2

2

4complex

plane trajectories

Im(ω

)10

4

(d)

2 4|ΔωI| 1040

-2

0

2

4

6

Im(ω

)10

4

Compatibilitywith plasmonics(fixed losses)


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

Ampl

itude

(a.u

.)Ph

ase

loss modegain mode

(d)

(e)

0 5 10 15n°4

0 5 10 150

0.5

1

-101

n°5

0 5 10 150

0.51

-101

n°6

Only modes#4and#5gotobroken sym


Fieldevolution with PT‐sym/just‐sym stripes

……

PT‐sym P‐only‐sym4 gainlobes

P‐only‐sym4 loss lobes


OUTLINE



Transverse PT symmetry with static disorder Can one apply « some transverse k » to a WG ? The medium is the message (?) : Transverse k-filters Connection with random matrix framework ? Longitudinal adiabaticity and (lack of) reciprocity



modes,m,n

Dropmodem

PT‐symmetricPotential

•Features : Static shortrange« noise »(disorder) Nonuniform profile Essentially we imposeatransversemomentum (kz)DISTRIBUTION

z

modes,m,n

kz

Re(ε),Im(ε)

(barrier)

Dropguide

(onmainguide)

Howperfect should aPTpotential be?


• •Procedure• • • Choose kz profile(e.g.Gaussian around kzc)

•During gain/loss scanGenerate random kdistrib.Solve EigenvaluesComparetonoPTreferenceForasubset ofthescan:

‐ Find eingenvectors‐ Launch 1by1no‐PTmodesatentrance‐ Monitorspecific responses atWGlateral positions

End;•End;

kz

Re(ε), Im(ε)

kz,c0

-kz,c

modes,m,n

Transversespatialfrequency spectrum

Fixed+randomcontributions(128‐vector)


Principle of[chirp+noise]

Scanaround athreshold (chirp),butinnoisy conditions

Canreveal morethan chirp ornoisealone

Example :percolationtreshold quest (1985,Rossoetal.PRB)

gradient

randomnessStats in frontier region is richerthan non-gradient (plain ergodic) stats


Example of[chirp+noise]

~12%chrp +3%noise

0.94

1.06

Gain/loss,PT‐part(odd)

(even orodd)

Real‐space profileis theFTofthedesired kprofile!


0 50 100 150 200 250 300 350

10-3

0

1

2

3

4

5

6

7

8

<abs(Im[ω])>

arparam=0.001 fixedparam=0.03; kc=20;dk=1;AR=diag((2*jtry/Ntry)*abs(kappa)*(arparam*randff+fixedparam*fixedodd));

kz,c = 20

3%random contributionvs.PTpart

Eigenvalues in[chirp+noise]scan


Amplified field of kc mode at kc(PT)=20mode#

Fieldsin[chirp+noise]scan

0.94 Chirp scan1.06

20dB

10dB

0dB

‐10dB

30

20

10

0

Intensity inwaveguide1.06

0.940

positionx inwaveguide 128

Chirpscan

•We get therightmodeset,20dBover(WGlength/noisecompromise)

•Itlookssensibletokeep PT‐sym profilelarger onside•Inpractice,assessment ofaveraging should notbe overlooked


Defining channels

0.94 Chirp scan1.06

30 dB

Intensity (dB)

Modenumber

‘Channelresponse’:lookatmodesinaggregates

20

0

30

10

40Modenumber


Intensity (dB)

Modenumber

Demultiplexing fourchannels

Channels taken atconstantlength butvarying PTgainloss parameter

g=2 g=3 g=3.75 g=3.75

0 20 40 60

-10

0

10

20

10 30 50


0 20 40 60

-10

0

10

20

30

40

Comparison ofPTandnonPT

PTbringssmoother response +flat‐top

P only / large gain-loss

P only / medium gain-loss

PT / large gain-loss

Intensity (dB)

Intensity (dB)

Intensity (dB)

Modenumber

Modenumber


Dispersion Orderedspacing

distribution

Momentum #number inseries

kz distribution

Spacinghistogram

(loglog)10-4 10-3 10-2

100

101

102

103

104

x10-3

1 2 3 4 5 6 7 80

500

1000

1500

2000

2500

3000

(linlin)

Related issue:level spacing inPT‐symmetric random systems


Adiabaticity and(lack of)reciprocity

modes,m,n

Dropmodemz

modes,m,n ???

(barrier)

Dropguide

AdiabaticPT‐symmetric regionwith given sign ofodd

partofIm(ε)

(Anti)AdiabaticPT‐symmetric region

with oppositesign ofoddpartofIm(ε)

PTsign :+ ??

PTsign :–


Length along wg

Length along wg

Mode number

Mode number

dBLocal eigenvalues

PTsign :+

PTsign :–

Adiabatic systemintensity response

Overall : better behaviour than abrupt PT potential


Adiabatic systemfield patterns

PTsign+

PTsign–

Mode 1 Mode 5 Mode 9 Mode 13

Mode 17 Mode 21 Mode 25 Mode 29

PTsign + PT Sign –


Other modes

Apparently :•PT‐symmetry =goodstarting point•Notrivialway (inlinear optics)toneutralize other modes’fate. Need forfull‐fledged optimization ?


CONCLUSION

PT–symmetry and multimode systems

Special interest of isolating gain of a few modes

Industrial need of multimode management

Noise more critical around exceptional point (no big suprise)

20 dB discrimination seems possible

Adiabatic evolution for good operation

Issue 1 : noise and level spacing around EP Issue 2 : Full « MIMO » modal sieve not so obvious

(partial success of « reciprocity recipe »)

Documents

H. Benisty A. Lupu2 and A. Degiron 2« Non-hermitian photonics in complex media », Crete, 15-18 June, H. Benisty et al. 1 Multimode waveguide modal management with PT‐ symmetry