Second-order stopband and leaky-mode resonator

(1)

Second-order stopband and leaky-mode resonator

Seok 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

(2)

Topological insulator laser: Theory

Gal Harari, et al. (Physics Department and Solid State Institute, Technion–Israel Institute of Technology, Haifa 32000, Israel Science 359, 4003 (2018) 16 March 2018

Non-Hermitian Exceptional Point

(NH-EP)

Topological Exceptional Point

(TP-EP)

Dirac-Point

연구목적 (장기)

(3)

연구목적 (중기)

Science 359, 1009–1012 (2018) 2 March 2018

We theoretically propose and experimentally demonstrate a bulk Fermi arc that develops from non-Hermitian radiative losses in an open system of photonic crystal slabs

second-order stop band.

We connect the fields of topological photonics, non-Hermitian physics, and singular optics.

354 | NATURE | VOL 525 | 17 SEPTEMBER 2015

second-order stop band.

(4)

연구목적 (단기)

Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals

41, 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

(5)

Bragg reflector and Resonance grating

(6)

Bragg diffraction

Bragg Reflection

Bragg Diffraction

(7)

Dispersion relation (photonic band structure)

Photonic Bandgap Formation

Band-edges Reduced (1

^st

Brillouin zone) band structure

(8)

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 I I

F

₁



 n

_c

n

_s

F

₂



M 

n

_h

d 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

(9)

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

(10)

Second-Order Distributed Feedback (DFB) Lasers with Mode Selection Provided by First-Order Radiation Losses

RUDOLF F. KAZARINOV AND CHARLES H. HENRY QE-21, 144 (1985)

IEEE J QE

Thus, the mode related to the gap edge with no radiation loss is favored for a DFB laser.

The dielectric function of the waveguide The grating layer

(Second-order) grating vector (K

₀

) = propagation vector

For a grating with reflection symmetry:

x z Stop bands in Symmetric profiles

the fundamental transverse mode for the waveguide without the periodic grating, with real dielectric function

’_O(x), propagation vector K₀. For the first-order stop band:

there are two edge modes and the inherent problem of DFB design is to suppress one of the modes.

For the second-order stop band:

the second-order diffraction provides Bragg reflection

the first-order diffraction causes 90" radiation losses.

The 90" radiation is a superposition of contributions from the two oppositely traveling waves which form the resonator mode.

The phase difference of the two contributions to 90" scattering depends upon photon energy relative to the gap in such a way that

At one edge of the gap the phase difference is 180" and the radiation cancels, At the other edge the phase difference is 0" and the radiation loss is nhanced.

radiating wave in first-order diffraction forward propagating backward propagating

In second-order diffraction

(11)

Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses

QE-21, 144 (1985)

IEEE J QE

The wave equation is given by

Setting the sum of terms with the same exp (imK_oz) (m = 0, +1, -1) to zero,

By introducing Green's function G(x, y) satisfying (G ~ Impulse response function)

The sum of the responses from the source points (y)

Radiation loss cancels when A + B = 0 (B = -A).

h₁is the coupling coefficient arising from two first-order diffractions h₂is the coupling coefficient arising from a second-order diffraction.

The imaginary part to the dielectric constant

Stop bands in Symmetric (reflection or inverse symmetric) profiles

x z

(2)

(3) (1)

(1) 

satisfies

(2) & (3) the dispersion relation,

relating changes in wavevector, changes in angular frequency (4)

(5) radiating wave

in first-order diffraction forward propagating backward propagating

In second-order diffraction

(v_g= __)

(12)

“Coupled Wave Analysis of DFB and DBR Lasers”

QE-13, 134 (1977) (참고) IEEE J QE,

₀= (n₀–i/k₀)²

we assume

forward propagating

backward in second-order diffraction

the mode gain/loss

(v_g) the deviation from the Bragg condition

(v_g= _)

(13)

Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses

QE-21, 144 (1985)

IEEE J QE

By substitution at k = 0 we have standing waves with zero group velocity.

If we neglect absorption and radiation losses,

The dispersion spectrum is

at k = 0 (or, |K| = K₀)

When losses are present, at k = 0 (Band edges),

The dispersion spectrum is The dispersion relation is

The mode field is the sum of two oppositely propagating Bloch waves

In general, the solution has two equal and opposite values of k

No radiation loss

Radiation loss

Stop bands in Symmetric profiles

(4)

(5)

(6)

(14)

Band gaps and leaky-wave effects in resonant photonic- crystal waveguides

Y. Ding and R. Magnusson

15, 680 (2007)

Opt Express

second stop band

The dispersion relation of the second-order band:

Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses QE-21, 144 (1985)

= 0

  k 

The grating layer

_m(x)

_m(x) = _m

For a binary grating, are constant.

h3 ~ the group velocity

Or, in a different form,

(15)

Band gaps and leaky-wave effects in resonant photonic- crystal waveguides

15, 680 (2007)

Opt Express

In the case when h₁=0 (the nonleaky band edge of the second-order band)

 real prop; k (w) complex freq. >

k  real freq;  complex prop. >

 k real freq;  complex prop.

 real prop; k  complex freq.

 propagation constant has an imaginary part

 k

no solution for frequency

In the case when h₁is not zero (the leaky band edge)

At frequency the upper band edge

h₁has no effect

Nonleaky band edge

approximately

At frequency

At frequencies far from the band edges

Leaky edge

(1) Band structure in real frequency

(16)

Band gaps and leaky-wave effects in resonant photonic- crystal waveguides

15, 680 (2007)

Opt Express

At frequency the upper band edge

h₁has no effect

approximately

At frequency

Leaky edge

(1) Band structure in real frequency (2) Band structure in real propagation constant

 k complex frequency,  real

 k real frequency,  complex

At frequency

(17)

Band gaps and leaky-wave effects in resonant photonic- crystal waveguides

15, 680 (2007)

Opt Express

(1) Band structure in real frequency (2) Band structure in real propagation constant

 k real frequency,  complex

 The dispersion plots in complex frequency give a more clearly defined band gap.

(18)

Band gaps and leaky-wave effects in resonant photonic- crystal waveguides

15, 680 (2007)

Opt Express

Phase matching condition to excite a resonance with a plane wave incident at angle θ,

The representation in complex frequency delivers the real propagation constant directly and matches the results from the diffraction computations.

_Rcurve (real frequency) and the locations of GMRs

Not correct Correct

GMR reflectance

_Rcurve (complex frequency) and the locations of GMRs

(19)

Use of nondegenerate resonant leaky modes to fashion diverse optical spectra

Second stopband characteristics

Opt Express

Asymmetric profile

TE₀mode

TE₁mode

n_s=1.48 n_h=2.05 n_avg=2.0

n_c=1.0 =0.3 m d=0.14 m F=0.33

n_s=1.48 n_h=2.05 n_avg=2.0

n_c=1.0 =0.3 m d=0.14 m F₁=0.2 F₂=0.13 M=0.5

TE₀mode

_ _

_

d/=0.47

n

_s

=1.48 n

avg

=2.445

n

_c

=1.0 d/=0.65

Asymmetric profile

Possible interaction between GMRs #2 & 3 as Δε increases.

Since each GMR is associated with 100% reflection, placing

two GMRs near each other opens the possibility of a flat

reflection band.

(20)

Use of nondegenerate resonant leaky modes to fashion diverse optical spectra

Opt Express

n_s=1.48 n_h=3.48

n_avg=2.445

n_c=1.0 =1.0 m d=0.67 m F₁=0.397 F₂=0.051 M=0.5

Example 1: Bandstop filter with narrow flattop

~ –35dB transmission dip with bandwidth about 2nm

by interaction of the differentiated TE0 and TE1 modes

n_s=1.48

n_h=2.8 n_avg=2.445 n_c=1.0

n_h=2.8 n_avg=2.445

n_h=3.48 n_avg=2.445

n_s=1.48 n_h=3.48

n_avg=2.45

n_c=1.0 =1.0 m d=0.45 m F₁=0.35 F₂=0.1 M=0.52

Example 2: Bandstop filter with wide flattop

~150 nm with central wavelength near λ=1.75μm

Example 3: Bandpass filter

n_s=1.48 n_h=3.48

n_avg=2.667

n_c=1.0 =1.0 m d=0.35 m F₁=0.5 F₂=0.05 M=0.55

the peak is due to the asymmetrical lineshape associated with GMR#2.

(21)

Use of nondegenerate resonant leaky modes to fashion diverse optical spectra

Opt Express

Asymmetric profile

TE₀mode

TE₁mode

_

n

_s

=1.48 n

avg

=2.445

n

_c

=1.0 d/=0.65

Possible interaction between GMRs #2 & 3 as Δε increases.

Since each GMR is associated with 100% reflection, placing two GMRs near each other opens the possibility of a flat reflection band.

n_s=1.48 n_h=3.48

n_avg=2.45

n_c=1.0 =1.0 m d=0.45 m F₁=0.35 F₂=0.1 M=0.52

Example 2: Bandstop filter with wide flattop

~150 nm with central wavelength near λ=1.75μm

Example 3: Bandpass filter

n_s=1.48 n_h=3.48

n_avg=2.667

n_c=1.0 =1.0 m d=0.35 m F₁=0.5 F₂=0.05 M=0.55

the peak is due to the asymmetrical lineshape associated with GMR#2.

(22)

Resonant leaky-mode spectral-band engineering and device applications

12, 5661 (2004)

Opt Express

Second stopband characteristics

Bandpass filters for TE polarization

n_s=1.48 n_h=3.48 n_low=1.0

n_c=1.0 =1.0 m d=0.7 m F₁=0.1 F₂=0.35 M=0.35

TE₀mode

TE₁mode

_

Two TE

^1,1

modes (GMR#3, GMR#4)

The transmission peak is provided by the asymmetrical lineshape of TE

_1,1

at the upper stopband edge which generates

the corresponding minima in the η

_R

curve near λ=1.65 μm.

FWHM passband of 3.5 nm

n_h=3.48 n_h=2.8 GMR#4

(shorter wavelength) (upper bandedge)

GMR#3 (longer wavelength)

(lower bandedge)

GMR#4

GMR#3

Mixed-1 GMR#4 (TE_1,1) GMR#5 (TE_1,2)

n_h=3.48 n_h=3.48

for different modulations with the effective index of the waveguide kept constant

Symmetric (S) mode Asymmetric (A) mode

Symmetric (S) mode?

Mixed-1 GMR#4 (TE_1,1) GMR#5 (TE_1,2) Mixed-2

GMR#4 (TE_1,1) GMR#5 (TE_1,2)

GMR#5 x x^GMR#6 TE₂mode?

Mixed-2 GMR#2 (TE_1,0) GMR#3 (TE_1,1)

(23)

Resonant leaky-mode spectral-band engineering and device applications

12, 5661 (2004)

Opt Express

Second stopband characteristics

Bandpass filters for TM polarization

n_s=1.48 n_h=3.48 n_low=1.0

n_c=1.0 =1.0 m d=0.81 m F₁=0.25 F₂=0.5 M=0.36

Mixed-1 GMR#4 (TM_1,1) GMR#5 (TM_1,2)

Mixed-2 GMR#2 (TM_1,0) GMR#3 (TM_1,1)

(shorter wavelength) (upper bandedge)

(longer wavelength) (lower bandedge) Symmetric (S) mode?

Symmetric (S) mode?

(Mirror) Wideband reflector for TE polarization

n_s=1.48 n_h=3.48 n_low=1.0

n_c=1.0 =1.0 m d=0.49 m F₁=0.075 F₂=0.375 M=0.45

symmetric profile

Asymmetric profile

the profile is symmetric in this example, only one GMR will exist for each stopband.

Since the total effective fill factor is less than 0.5 (F=0.45),

the GMR appears at the lower bandedgeand concentrates in the high-ε region.

GMR#1 (TE_1,0) GMR#3 (TE_1,1)

GMR#3 (TE_1,1) GMR#1 (TE_1,0)

No GMR#2, GMR#4 (low-e region) since symmetry

(24)

Resonant leaky-mode spectral-band engineering and device applications

12, 5661 (2004)

Opt Express

Second stopband characteristics

(Mirror) Wideband reflector for TM polarization

n_s=1.48 n_h=3.48 n_low=1.0

n_c=1.0 =1.0 m d=0.81 m F₁=0.25 F₂=0.5 M=0.375

symmetric profile

the profile is symmetric in this example, only one GMR will exist for each stopband.

Since the total effective fill factor is less than 0.5 (F=0.45),

the GMR appears at the lower bandedgeand concentrates in the high-ε region.

GMR#1 (TM_1,0) GMR#3 (TM_1,1)

GMR#3 (TM_1,1)

GMR#1 (TM_1,0)

Mixed GMR#4 (TM_1,1) GMR#5 (TM_1,2)

GMR#1 (TM_1,0) GMR#3 (TM_1,1)

Mixed-1 (short wavelength)

Mixed-2 (long wavelength)

All symmetric (S)

(25)

Resonant leaky-mode spectral-band engineering and device applications

12, 5661 (2004)

Opt Express

Second stopband characteristics

Polarizers

n_s=1.48 n_h=3.48 n_low=1.0

n_c=1.0 =0.84 m d=0.504 m F₁=0.025 F₂=0.675 M=0.05

Asymmetric profile

GMR#2 (TM_1,0) GMR#3 (TE_1,1)

GMR#4 (TE_1,1)

GMR#2 (TE_1,0)

GMR#3 (TM_1,1) GMR#4 (TM_1,1)

GMR#4 (TM_1,1) (short wavelength)

GMR#3 (TM_1,1) (long wavelength)

TE reflectance TM transmittance

(26)

Resonant leaky-mode spectral-band engineering and device applications

12, 5661 (2004)

Opt Express

Second stopband characteristics

Polarization independent element

n_s=1.52 n_h=3.48 n_low=1.0

n_c=1.0 =0.84 m d=0.6 m F₁=0.435 F₂=0.225 M=0.5

Asymmetric profile

TE reflectance TM reflectance

TE TM Transmittance TE TM reflectance

n_h=2.8 n_h=2.8

n_h=3.48

Antireflection element Asymmetric profile

n_s=1.48 n_h=3.48 n_low=1.0

n_c=1.0 =1.0 m d=0.85 m F₁=0.2 F₂=0.25 M=0.375

TM reflectance

(27)

Dispersion Engineering with Leaky-Mode Resonant Photonic Lattices

15, 680 (2007)

Opt Express

Time-delay elements based on leaky-mode resonance for optical buffers, delay lines, and switches.

A transform-limited TE-polarized Gaussian pulse is

the incident Gaussian pulse is decomposed into its monochromatic Fourier components (plane waves) These discrete monochromatic components are then treated independently by our established RCWA analysis technique

The time delay () and delay dispersion (D) are

()  dependent phase shift after reflection or transmission

 = 979 nm, d = 465 nm, n_H= 3.48, n_S= 1.48, and n_L= n_inc= 1.0 [F₁, F₂, F₃, F₄] = [0.071, 0.275, 0.399, 0.255]

TE incidence

delay of ~10 ps

(28)

Dispersion Engineering with Leaky-Mode Resonant Photonic Lattices

15, 680 (2007)

Opt Express

 = 1103.9 nm, d = 432.2 nm, d_cavity= 2000 nm n_H= 3.48, n_S= 1.48, and n_L= n_inc= 1.0 [F₁, F₂, F₃, F₄] = [0.0626, 0.3013, 0.4576, 0.1785]

an average delay of ~7 ps

(29)

Leaky-mode resonant reflectors with extreme bandwidths

Mehrdad Shokooh-Saremi and R. Magnusson 35, 1121 (2010)

Opt Lett

 = 846.4 nm,

n_H= 3.48, n_S= 1.48, and n_L= n_inc= 1.0 d1=375 nm, d2=175 nm, d3=375 nm

F11=F31 = 0.283, F21=1.0

TE incidence

TM incidence

 = 882.4 nm,

n_H= 3.48, n_S= 1.48, and n_L= n_inc= 1.0 d1=572 nm, d2=455 nm,d3=572 nm

F11=F21=F31=0.417

n

_GE

= 4.0

multilevel resonant leaky-mode structures

(30)

Guided-mode resonant wave plates

Opt Lett

 = 786.8 nm,

n_H= 3.48, n_S= 1.48, and n_L= n_inc= 1.0 d1 =525:3 nm, d2 = 624:6 nm

F=0.2665

Half-wave plate (Si) Fabrication tolerance

deviation (5:0%) in silicon thickness

deviation in grating fill factor (F)

 The retarder is rather sensitive to deviations in Si thickness.

(31)

Guided-mode resonant wave plates

Opt Lett

Half-wave plate (Si + HfO ₂ )

HfO₂ Si

 = 780.9 nm,

n_Si= 3.48, n_Hf)2= 2.0, n_S= 1.48, and n_L= n_inc= 1.0 d_HfO2=1086.2 nm, d_Si= 375.3 nm

F=0.4876

Quarter-wave plate (Si + HfO2)

 = 878.9 nm,

n_Si= 3.48, n_Hf)2= 2.0, n_S= 1.48, and n_L= n_inc= 1.0 d_HfO2=586.1 nm, d_Si= 413.8 nm

F=0.3072

HfO₂ Si

(32)

Tunable guided-mode resonances in coupled gratings

Hahn Young Song, Sangin Kim*, and Robert Magnusson 17, 23544 (2009)

Opt Express

A generalized theoretical analysis on the tunable characteristics of the GMRs in coupled gratings

() Direct (evanescent) coupled

() Radiation coupled via the leakage wave (outgoing wave)

Let’s investigate the general resonance characteristics of the coupled resonator by using the coupled-mode theory

the incoming wave amplitudes

the outgoing wave amplitudes

Radiation coupling coefficient



Direct (evanescent) coupling coefficient

_r (radiation coupling coefficient)

_i (evanescent coupling coefficient)

(33)

Tunable guided-mode resonances in coupled gratings

Opt Express

  phase retardation

after propagation (distance d)

i



Because of energy conservation and time reversal symmetry,  is real

Define the amplitudes of the super-modes with even and odd symmetries

Reflection coefficient (when S

₊₄

= 0)

(34)

Tunable guided-mode resonances in coupled gratings

Opt Express

Reflection coefficient (when S

₊₄

= 0)

Even Odd

Odd Even

Even Odd

For ( )

the odd super-mode gets broader and the even super-mode peak gets sharper.

For a strong direct evanescent coupling regime, ( ,  ), the resonance frequency tuning is dominantly governed by 

the phase retardation  does not change the resonance peaks much but mainly affects the linewidths of the peaks.

For (



0) such that evanescent coupling is negligible, the phase retardation (  ) plays an important role

Reflection dip (transmission peak) occurs at () No Odd

() No Even

s ( 0  P/2) means the phase retardation  . the Zak phase??

(35)

Tunable guided-mode resonances in coupled gratings

Opt Express

In strong evanescent coupling regime

50 nm thick SiN (n = 1.85)

P = 800 nm

F = 0.5

330 nm thick chalcogenide glass (n = 2.38)

For a single grating normally incidence (TE).

a narrow linewidth of 1.5 nm (λo = 1599.2 nm)

( ) the Odd modes get sharp

( ) the Even modes get sharp

Odd Even

d ( 0  500 nm) means the evanescent coupling strength  .

s ( 0  P/2) means the phase retardation  . the Zak phase??

Grating (1.85)

Air (1.0) (2.38)

(1.5) (1.5)

Even

Odd

d = 0

(36)

Tunable guided-mode resonances in coupled gratings

Opt Express

In strong evanescent coupling regime

( ) the Odd modes get sharp

( ) the Even modes get sharp

Odd Even

s ( P/2) means the phase retardation  . the even width  0

the effect of the gratings on the field is very small.

s ( P/4) means the phase retardation  .

a very sharp reflection dip (a transmission peak) is observed near the even mode peak for d < 100 nm,

The dip results from a destructive interference between two nondegenerate GMRs associated with the even mode. In an asymmetric waveguide grating, both edges of the second stop band support GMRs interface mode?? CHECK MODE PROFILE!!

(37)

Tunable guided-mode resonances in coupled gratings

Opt Express

In negligible evanescent coupling regime

The calculated spectra are independent of the horizontal alignment of two grating since free space propagation is a dominant coupling mechanism.

TM polarization

(38)

Experimental and Theoretical Demonstration of Resonant Leaky- Mode in Grating Waveguide Structure with A Flattened Passband

National Central University, Taiwan 46, 5431 (2007)

JJAP

Lateral shift: 0.5  f. factors: 0.12 and 0.38 Poly-Si (3.48)

SiO2 (1.46)

 = 700 nm d_g= 400 nm d_w= 100 nm

(39)

Evolution of modes of Fabry–Perot cavity based on photonic crystal guided-mode resonance mirrors

Pierre Pottier, Lina Shi, and Yves-Alain Peter, Ecole Polytechnique de Montréal 29, 2698 (2012)

JOSA B

Filled dots: high transmission peaks (>90%), empty dots: lower transmission peaks (<90%).

points () points ()

The modes of a Fabry–Perot cavity made of two-dimensional photonic crystal guided-mode resonance mirrors (Si3N4 ⁄air), and compare it with an ideal Fabry–Perot cavity and a cavity made of Bragg mirrors.

Evolution of modes of an ideal FP cavity (black dashed straight lines), an FP cavity made of Bragg mirrors (red curves), and an FP cavity made of PhC GMR mirrors (dots)

(40)

Tunable transmission filters based on double subwavelength periodic membrane structures with an air gap

Tian Sang1,2, Tuo Cai1, Shaohong Cai2 and ZhanshanWang, China 29, 2698 (2012)

JOSA B

GaAs (3.5) air (1.0)

 = 631.4 nm d₁= 280 nm d₂= 60 nm f.f = 0.18

S = 0.0 d_g= 80 nm d_g= 450 nm

S = 0.31, dg = 80 nm

1214.8 nm 1185.1 nm

1182.5 nm S = 0.0, dg = 80 nm

1196.3 nm

S = 0.31, dg = 450 nm

1098.8 nm Single GMR