GMR and Topological Photonics - Hanyangoptics.hanyang.ac.kr/~shsong/2-GMR and Optical Topology.pdf · GMR and Topological Photonics SeokHo Song (HYU) The first-order photonic stopband

GMR and Topological PhotonicsSeok Ho Song (HYU)

The first-order photonic stopband has widely been utilized for photonic crystal applications.

At higher-order stopbands including second-order one, the coupling between GMR (leaky) band-edge states and a continuum of radiation states from the higher-order stopband can generate an confined mode, which can guide through the topological interface (edge).

There may exist a strong relation between EP in NH photonics and DP in topological photonics.

“It is worth mentioning the potential of utilizing exceptional point singularities in optical scattering problems,where the coupling between discrete localized metastable states and a continuum of radiation states is concerned.Such optical scattering problems in general, be treated as non-Hermitian problems, for which a point of interest would be to explore the connection between radiation leakage and exceptional points emerging in the continuum.” (Mohammad-Ali Miri and Andrea Alù, Science 363, 42 (2019)

2019-02-28

second-order stop band (high contrast GMR)

first-order stop band (photonic crystal)

third-order stop band

연구목적 (단기)Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals41, 1644 (2016)

Optics Letters

there are strongly localized fields near the interface between X and Y’.The enhancement at the edge mode frequencies are about 20 to 30.

The enhancement can increase exponentially with increasing number of units. This result suggests that the interface region is suitable for inserting single or few layers of thin or 2-D materials for enhancing frequency upconversion optical processes.

Plane wave incidence

Dispersion relation (photonic band structure)Photonic Bandgap Formation

Band-edges Reduced (1st Brillouin zone) band structure

Guided-mode resonance (GMR) from symmetric and asymmetric grating profiles

z

x

F

T y p e I

(b) Grating with asymmetric profile

T y p e II

F 1

n c

n s F 2

M

dn h n l

(a) Grating with symmetric profile

Details: Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Optics Express, May 3, 2004

For symmetrical grating (Type I):

leaky mode in phase

For asymmetrical grating (Type II):

leaky mode in phase

Non-leaky mode (out of phase, )

Stop bands in Symmetric and Asymmetric profiles

leaky mode in phase

Use of nondegenerate resonant leaky modes to fashion diverse optical spectra

Y. Ding and R. Magnusson

12, 1885 (2004)

Opt Express

Band crossing (Dirac cones), Band flips,Flatbands

in photonic lattices

Band flips and bound-state transitions in leaky-mode photonic latticesSun-Goo Lee and Robert Magnusson99, 045304 (2019)

PR B

It is possible to control the width of the 2nd-order band gap by lattice design. In particular, as a modal band closes, there results a degenerate state (band crossing) the degenerate point executes a band flip.

Non-leaky modes via bound-states in the continuum (BIC)

leaky modes via guided-mode resonances (GMRs)

For symmetrical grating (Type I):

Before band flip

(Infinite thickness) 1D non-leaky photonic crystal

Band closure After band flip

(Finite thickness) 1D leaky photonic crystal

Before band flip Near band closure After band flip


PR B

= (/c)/K = k0/K normalized frequency

In order to calculate the three coupling coefficients h0, h1, and h2. the grating layer is treated as a homogeneous waveguide

The mode profiles are given by

wavevector components along the x-direction

by normalization

Semi-analytical calculation of dispersion relations

= Bragg frequency under vanishing index modulation

(= h3)The Green’s functions are given by

wavevector components along the x-direction

transmission coefficient from the regions i to j

reflection coefficient from the regions i to j


PR B

Semi-analytical calculation of dispersion relations

For the 1D symmetric lattice,

FDTD (harmonic) KH (Semi-analytic)FDTD (full)

Note that the sign of h2,

BIC

GMR

When > 0.5 2 < 0 h2 < 0

The size of the 2nd-order band gap is

When < 0.5 2 > 0 h2 > 0 can be gap closing

can be wide-gap opening

At the 2nd-order band gap closure with

The frequency is fully degenerate (not coalescent) at

Exceptional point

Symmetry Breaking in Photonic Crystals: On-Demand Dispersion from Flatband to Dirac Cones H. S. Nguyen ….., Universit´e de Lyon

120, 066102 (2018)PRL

Fundamental zero-order waves of the two noncorrugated waveguide structures

diffractive coupling rates between forward and backward

diffractive coupling of the evanescent part of the other grating

first-order diffraction phase shift due to the offset

Evanescent coupling rate between waves of different gratings but propagating in the same direction

group velocities in the noncorrugated waveguiding structures

Consider the first stopband first-order diffractive coupling processes between backward and forward wave components are most effective. second-order terms (anti-diagonal terms) that can be neglected.

(1) Fishbone structure

Since the two sub-gratings are identical,

From the four Fundamental zero-order waves,The weights of even and odd component in each eigenstate are

Four eigenstates |n> with n=1,2,3 and 4.

A double-period design perturbation with a perturbation magnitude α

120, 066102 (2018)PRL


Dirac cones

two Dirac points

Group velocities at the two Dirac points

The formation of Dirac cones are not accidental (i.e. independent on v0, U, and Vf, thus independent of the geometry and material of the sub-gratings). The only condition is the phase-shift, corresponding to an offset =0.5

At the Dirac points, the weight ratio between the odd and even component is

even and odd components are equally balanced (i.e. fully hybridized) for each Dirac point.

(2) Comb structure

the lower sub-grating of the Fishbone is non-corrugated,

Four eigenstates are

Dirac cones


120, 066102 (2018)PRL

Reflectivity(TE-pol)

high index mode (TE0) Vertical Even symm

low index mode (TE0) Vertical Even symm

first excited mode (TE1)of high index

Vertical Odd symm

At > 0, strong coupling between Even (C) and Odd (B) Tuning the vertical symmetry breaking by

At = 0.2, A flatband at Γ

At = 0.5, Dirac cones at Γ


/a = 0

(참고: coupling TE0, TE1; and GMR modes

GMR #2

GMR #1

GMR #3

Reflectivity (TE-pol)


Flatbandat Γ

Dirac coneat Γ

120, 066102 (2018)PRL

(이기영계산) Fishbone structure

/a = 0

/a = 0

Reflectivity (TE-pol)


120, 066102 (2018)PRL

A double-period design perturbation approach with a perturbation magnitude α

the integration of active materials in these designs, for example, by placing quantum wells in the noncorrugated part of a comb structure, or by depositing monolayers of 2D material on top of the structure via exfoliation.

the active layer will be unpatterned—an ideal configuration to obtain the strong coupling regime with fragile 2D materials the possibility to use multivalley dispersion in the strong coupling regime to obtain spontaneous momentum symmetry breaking

and two-mode squeezing, as well as the observation of the Josephson effect in momentum space

Experiment


(2) Comb structure

Flatbandat Γ

At = 0.5, Dirac cones at Γ

Dirac coneat Γ

Flatbandat Γ

88, 245124 (2013)PR B

천상모 교수님 ???

Solitons in one-dimensional three-band model with a central flat bandGyungchoon Go, Kyeong Tae Kang, and Jung Hoon Han, Sungkyunkwan University

Topological interface modes in 1-dim. photonic lattices

Surface impedance Zak phase (1-dim. Berry phase)

Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems4, 021017 (2014)

PR X

Impedances and Zak phases

Impedance ZA,B (i) of materials oii

i

i

ii nH

E

1

Inside a band gap, ZSR is a pure imaginary number

The condition for the presence of an interface state

zm zd

m d

k k

the surface impedances on the two sides are opposite in sign

The sign of the surface impedance for frequencies inside a band gap

is, in fact, determined by the geometrical phase (Zak phase) of the

bulk bands.

Let’s derive a relation between the surface impedance and the Zak phase

Surface impedance ZSR of the semi-infinite PC

Let us now put another PC,

ZSL ZSR

Meng Xiao, Z. Q. Zhang, and C. T. Chan

(Z0: the vacuum impedance)

4, 021017 (2014)

PR X

ZAK PHASE OF EACH BAND

Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems

x

z

unit cell

The periodic-in-cell part of the Bloch E field of a state on the nth band with wave vector q

Berry connection

The 1D system with inversion symmetry always has two inversion centers (A or B)

the Zak phase is quantized at either 0 or ,if the origin is chosen to be one of the inversion center.

ZAK PHASE = 0 or Show that if one isolated band (excluding the 0th band) contains the frequency point at which (if we set the origin of the system at the center of slab A).

the Zak phase of this band must be . Otherwise, the Zak phase of this band must be

( in band 7)

When εa is decreased from 4, we need to increase the value of da in order to keep nada + nbdb unchanged. Thus, the value of nbdb is reduced accordingly, which in turn implies

( in band 6)

4, 021017 (2014)

PR XC. Zak phase and the symmetry properties of the edge states


As an example, let us focus on the 6th and 7th bands in which the Zak phases change by π.

Before crossing After crossing

(N, R) Even functions E (z) not affected by the band crossing

(L P) After the band crossing,Odd functions E (z) Even Function

(M Q) After the band crossing,Even functions E (z) Odd Function

(M, N) Same symm. Zak phase = 0

(Q, R) diff. symmetry Zak phase =


Odd Even

OddEven

Even Even

4, 021017 (2014)

PR X

D. The sign of impedance and the symmetry properties of the edge states


For (L & Q) Antisymmetric (A) state E = 0 at the boundary A perfect electric conductor-like boundary

condition Reflection must be r = -1 = exp(i)

For (M & P) Symmetric (S) state E = maximum at the boundary A perfect magnetic conductor-like boundary

condition Reflection must be r = 1 = exp(i)

For a gap with state A at the lower edge (point Q),The function ς has a value 0 at the lower edge and decreasesmonotonically to −∞ as the upper edge is approached.

For a gap with state S at the lower edge (point M),The function ς has a value ∞ at the lower edge and decreasesmonotonically to 0 as the upper edge is approached.

Negative when the lower edge is type A.

Positive when the lower edge is type S.

If two states at the lower edges of the common gap have different types, an interface state must exist inside the gap.

The condition for interface states

4, 021017 (2014)

PR X

Zak phase and the sign of impedance


A relation between the surface impedance of the PC in the nth gap and the sum of Zak phases of all bands below the nth gap

l is the number of crossing points under the nth gap

(m =0)

(n = 1, l =0, Sum_Zak = )

(n = 2, l =0, Sum_Zak = )

(n = 3, l =0, Sum_Zak = 2)

(n = 4, l =0, Sum_Zak = 2)

(n = 5, l =0, Sum_Zak = 3)

(n = 6, l =0, Sum_Zak = 3)

(n = 8, l =1, Sum_Zak = 4)


(n = 7,l = 0, Z= 3)

(n = 7,l = 0, Z= 4)

4, 021017 (2014)

PR XChanging the sign of impedance by passing a topological transition point (Dirac point)

10 unit cells 10 unit cells

At crossing in 7th bandgapBefore crossing After crossing

The gap with (magenta)


The gap with (cyan)

Keep the condition of

The band crossing condition at (ex. m = 7)

Sign change inthe surface impedance in the 7th gap

Switching the Zak phase in passbands 6 and 7(0, )

DIRAC POINT

= (4/3)

= 7

= 5c/

Topological phase transition arising from band crossing in photonic systems

4, 021017 (2014)

PR X

A. Band crossing condition

the ratio of the optical path in A and B

the phase delay in A+B

the impedance mismatch

Band dispersion relation


passband bandgap

The sufficient condition for two bands to cross (either at zone center or zone boundary)

, which is always less than 1

The frequency at which sin(kbdb) = 0 must be in the pass band.

(l, m2: integer)

When (m1, m2: integer)

Near these frequencies,

The band has linear dispersion. Proof

Keeping to the lowest order of expansion of the band dispersion,

the degeneracy band point at

Suppose that (q1, ω1) is another band point NEAR (q0, ω0)

or

are small numbers.

(DIRAC POINTs)

When

bands will cross with linear dispersion.

4, 021017 (2014)

PR X

A. Band crossing condition

Meng Xiao, Z. Q. Zhang, and C. T. Chan


The necessary condition for two bands to cross

(1) The cross points of two bands could occur only at the boundary or the center of the BZ for 1D PC cases.

If two bands cross at points other than the center or boundary of the BZ, each frequency would have four corresponding Bloch vectors q.

This is not possible because there can be at most two values of q for each frequency.

(2) If one frequency satisfies the condition of

: Mid-gap frequency (or, Crossing frequency)

and

The frequency must be in the band gap if two bands do not cross.

If two bands cross at

Simultaneous multi-frequency topological edgemodes between one-dimensional photonic crystals41, 1644 (2016)

Optics Letters

If the nA of X is tuned just before topological phase transition, and the nC of Y is just after the transition, the combined structure (Fig. 1) will support a topological edge mode within this band crossing gap due to their π difference in

The band crossing condition

all the crossings will occur at every bandgap that opens at zone center.

Simultaneous multi-frequency topological edgemodes between one-dimensional photonic crystals41, 1644 (2016)

Optics Letters

there are strongly localized fields near the interface between X and Y’.The enhancement at the edge mode frequencies in Fig. 4 are about 20 to 30,

while the enhancement region has a two-dimensional (2-D) surfacearea of beam size of the incident light in plane wave approximation.

The enhancement can increase exponentially with increasing number of units. This result suggests that the interface region is suitable for inserting single or few layers of thin or 2-D materials for enhancing frequency upconversion optical processes.

Plane wave incidence

Measurement of the Zak phase of photonic bands through the interface states of a metasurface/photonic crystal

Qiang Wang,1 Meng Xiao,2 Hui Liu,1,* Shining Zhu,1 and C. T. Chan, Nanjing University

93, 041415(R) (2016)

Using a reflection spectrum measurement, we determined the existence of interface states in the gaps and then obtained the Zak phases.

PR B

nano

slits

etch

ed o

n a

silv

er fi

lm

HfO2

SiO

2If the two band edge states of the same band have the same symmetry, the Zak phase of this band is 0, otherwise, the Zak phase is π.

the determination of the symmetry types of the band edge states is quite difficult to implement experimentally

The Zak phase is related to the signs of the reflection phases,

the reflection phases of the (n−1)th gap and nth gap

we only need the sign, which greatly simplifies the measurement.

SA

The existence condition of an interface state is

The crossing points in band gaps 3 and 4. Or, from the reflection spectrum which is

presented as a dip inside the gap region.

(a) Between a silver film and a PC

x

z

Topological photonic crystal nanocavity laserYasutomo Ota1, Ryota Katsumi2, Katsuyuki Watanabe1, Satoshi Iwamoto1,2 & Yasuhiko Arakawa, University of Tokyo86, 1 (2018)

A topological PhC nanocavity with a near-diffraction-limited mode volume and its application to single-mode lasing. The topological origin of the nanocavity, formed at the interface between two topologically distinct PhCs, guarantees the existence of only one mode within its photonic bandgap.

COMMUN. PHY.

d1 + d2= 0.5a

d1 = 0.19 a d2 = 0.31 a

a = 270 nm

Near Dirac point

Documents

GMR and Topological Photonics - Hanyangoptics.hanyang.ac.kr/~shsong/2-GMR and Optical Topology.pdf · GMR and Topological Photonics SeokHo Song (HYU) The first-order photonic stopband