Hourglass nonlinear photonic crystal cavity for ultra-fast all-optical switching

Accepted Manuscript Title: Hourglass nonlinear photonic crystal cavity for ultra-fast all-optical switching Authors: Rebhi Sana, Monia Najjar PII: DOI: Reference:

S0030-4026(18)31925-9 https://doi.org/10.1016/j.ijleo.2018.12.001 IJLEO 62017

To appear in: Received date: Revised date: Accepted date:

16 August 2018 29 November 2018 1 December 2018

Please cite this article as: Sana R, Najjar M, Hourglass nonlinear photonic crystal cavity for ultra-fast all-optical switching, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Hourglass nonlinear photonic crystal cavity for ultra-fast alloptical switching

Rebhi Sana1, Monia Najjar1,2 1

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University of Tunis El Manar, National Engineering School of Tunis Communications Systems LR-99-ES21(LR-Sys’Com-ENIT), 1002, Tunisia 2 University of Tunis El Manar, Higher Institute of Computer2080, Ariana, Tunisia

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Abstract

In this paper, a new design of an ultra fast all optical switch based on a new photonic crystal cavity combined with Kerr effect is proposed and its performance is studied. Numerical

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methods such as plane wave expansion (PWE) and finite difference time domain (FDTD) are

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used to perform simulations and study the optical properties of the proposed switch.

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Switching from ON to OFF status occurred when the resonant wavelength is shifted due to

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Kerr effect. The simulation results show that switch has a high-speed response of ~15ps and

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since it has an ultra-small size of 273.24 μm2, it has a potential to be used in integrated optical-circuits.

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Keywords: All-optical switching; Photonic crystal cavity; Finite-Difference Time-Domain

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method; Optical Kerr effect.

1. Introduction

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In recent years, photonic crystals (PhCs) has been used for designing optical components on account of multiple features as compactness, ultra-high speed, electromagnetic wave controllability and its property of integration in optical circuits. Moreover, the wave guiding [1,2] , the wavelength selection [3,4] and the switching [5-7] properties of PhC made them very important to design diverse all optical devices. The combination of Kerr effect with PhC-

based resonators such as cavities [8-10] or ring resonators [11,12] is an ordinary mechanism used to design many optical logic gates [13-15], switches [5,6], decoders [16,17], encoders [18,19] and analog to digital converters [20,21]. The working mechanism of these structures is based on the fact that the resonant mode of PhC-resonators depends on the refractive index,

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which on its turn depends on the optical injected power All-optical switching is a key component in ultrafast communication and signal processing systems. In all-optical switching process, using nonlinear optical material, light

is be

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controlled by input intensity. The driving energy and the small size of photonic components are limited by the low confinement of light in a slight space and a weak light-matter

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interaction. Photonic crystals (PhCs), structures wherein refractive index changed

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periodically, are expected to overcome this limitation [22]. There are numerous methods to

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realize photonic crystal all-optical switches such as directional coupler structures [23-25],

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Mach–Zehnder interferometers [26-28] and resonators [27-29]. The main problem in directional coupler and Mach–Zehnder structures is their lengths, which is the most important

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drawback for its integration [25, 28].

The benefit of micro and nono-cavities is to constraint the light in a highly limited region.

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Consequently, it increases the interaction of the light with matter and reduces the device size.

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With the use of micro and nano-cavities for large Q-factor volume ratio (Q/V), all-optical nonlinear switch will exhibit significant reduction on switching energy [32,33]. In this work, we demonstrate that a In0,53Ga0,47As photonic crystal structure with nonlinear

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GaAs cavities coupled to input and output waveguides can operate as an ultra fast all-optical switch using Kerr effect with a good extinction ratio, insertion loss comparing to previous works. The current paper is structured as follows: in section 2, we present the nonlinear effects as well as the design of the proposed cavity structure. The simulation of the switching operation,

the calculation of transmission spectra and temporal behavior of the proposed cavity based switch is shown in section 3. Finally conclusion is drown in section 4.

2. Nonlinear effects and proposed cavity design The modification of material optical properties with presence of light power can engender

structures is governed by macroscopic Maxwell equations given by: 𝜕𝐷 𝜕𝑡

; ∇. 𝐵 = 0 ; ∇. 𝐸 = 𝜌

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𝜕𝐵

∇ × 𝐸 = − 𝜕𝑡 ; ∇ × 𝐻 = 𝐽 +

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optical nonlinearity phenomenon [15, 42,43]. The light propagation in photonic crystal

(1)

Here J is the current density and ρ is the free electron charge density. The four

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electromagnetic fields B, H, E and D are related between each other by the following

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expressions:

(2)

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𝐷 = 𝜀0 𝐸 + 𝑃

(3)

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𝐵 = 𝜇0 (𝐻 + 𝑀)

The polarization field is denoted by P and the magnetization field is denoted by M. The

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induced polarization of the medium inside a dielectric is represented by [34] (4)

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𝑃 = ɛ0 [1 + 𝜒 (1) 𝐸 + 𝜒 (2) 𝐸. 𝐸 + 𝜒 (3) 𝐸. 𝐸. 𝐸] + ⋯

Where 𝜒 (1) , 𝜒 (2) and 𝜒 (3) are the first, second and third order susceptibilities respectively.

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𝜒 (2) is only observed in non-centrosymmetric crystals while third-order susceptibility occurs for both centrosymmetric and non-centrosymmetric media and gives rise to Kerr nonlinearity. A monochromatic electric field E= 𝐸𝜔 cos ωt induce polarization of the medium [35] given by: 3

𝑃 ≈ 𝜀0 (𝜒 (1) + 4 𝜒 (3) |𝐸𝜔 |2 )𝐸𝜔 cos 𝜔𝑡

(5)

Considering the susceptibility as the sum of linear χ L and nonlinear χ NL components, the index of refraction is expressed by: 1

1

𝜒

𝑛 = (1 + 𝜒)2 = (1 + 𝜒𝐿 + 𝜒𝑁𝐿 )2 ≈ 𝑛0 (1 + 2𝑛𝑁𝐿2 ) = 𝑛0 + 0

3𝜒(3) 8𝑛0

× 𝐸𝜔 2 = 𝑛0 + 𝑛2 × 𝐼 (6)

Where n0 expressed by √𝜒𝐿 + 1 is the linear refractive index, n2 represents the second order

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refractive index and I is the light intensity. The electric field as well as displacement fields are

𝐷(𝑡) = 𝜀0 𝜀𝑟 𝐸(𝑡) = 𝜀0 𝑛2 𝐸(𝑡) = 𝜀0 (𝑛0 2 +

3 4

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currently related by: 𝜒 (3) |𝐸(𝑡)|2 ) 𝐸(𝑡)

(7)

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The most often used time-domain differential equation technique, in computational

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electrodynamics (CEM), is finite difference time domain method (FDTD) and it mostly helps

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in solving time dependent Maxwell’s equations setted in equation 1 with:

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𝐷 = 𝜀0 𝜀𝑟 𝐸

(9)

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𝐵 = µ0 µ𝑟 𝐻

(8)

As a perfectly matched layer (PML) is important on the FDTD simulation, electric fields

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should be normalized so that electric and magnetic fields have the same order of magnitude.

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Thus, the normalized fields are expressed by: ̃= 𝐷

𝐷

√ 𝜇0 𝜀 0

𝑐𝐷

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𝜇 𝐸̃ = √𝜇0 𝐸

(10)

(11)

0

Therefore, Maxwell’s equations with normalized electric fields become: ∇ × 𝐸̃ = −

𝜇𝑟 𝜕𝐵 𝑐

𝜕𝑡

(12)

̃ 1 𝜕𝐷

∇×𝐻 =

(13)

𝑐 𝜕𝑡

When deriving equations in the main FDTD algorithm, there will be fewer complications due to the fact that the above cited equations are independent of the material in use. The constitutional connection relating the E and the D-fields will be considered for the material

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properties given by the Kerr effect. Supposing that the z direction, of infinite extent (i.e. ∂/∂z= 0), is uniform and the material relative permeability ( µr ) is diagonal , equations in 8 and 9

𝜕𝐸̃𝑧 𝜕𝑥

𝜕𝐸̃𝑧 𝜕𝑦

𝜕𝐸̃𝑧

=

𝜕𝑦

=−

̃𝑧 1 𝜕𝐷

=𝑐

𝜇𝑦𝑦 𝜕𝐻𝑦 𝑐

𝜇𝑥𝑥 𝜕𝐻𝑥 𝑐

𝜕𝐸̃𝑥 𝜕𝑦

𝜕𝑡

𝜕𝑡

𝜕𝑡

U 𝐶𝑧𝐻

= 𝐶𝑦𝐸

= 𝐶𝑥𝐸

=−

𝜇𝑧𝑧 𝜕𝐻𝑧 𝑐

𝜕𝑡

(16)

= 𝐶𝑧𝐸

(17)

(18)

(19)

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−

𝜕𝐻𝑥

(15)

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𝜕𝑥

−

= 𝐶𝑥𝐻

(14)

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𝜕𝐻𝑦

𝜕𝑡

= 𝐶𝑦𝐻

M

̃𝑥 1 𝜕𝐷

=𝑐

𝜕𝑦

𝜕𝑥

𝜕𝑡

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=𝑐

𝜕𝑥

𝜕𝐻𝑧

−

̃𝑦 1 𝜕𝐷

𝜕𝐻𝑧

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−

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can be expanded as follows:

These equations match almost exactly to the TM mode and the TE mode respectively. In our work, the proposed photonic crystal switch is illustrated in figure 1. The designed

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structure is based on a square lattice two-dimensional photonic crystal formed by In0,53Ga0,47As rods of refractive index n=3.59 in air background . The lattice constant (a), the fill factor (r/a) and the length of the structure (L) are 551nm, 0.2 and 17μm, respectively.

The proposed structure exhibits two PBG for TE mode between the normalized frequencies 0.28< a/λ<0.43 and 0.72
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TE

TM

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1.0 0.8

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0.6 0.4

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Frequency (a/2c=a/)

1.2

0.0



M



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X

M

A

0.2

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Fig. 1 Photonic crystal band structure

To create the resonant cavity, we remove a 7 x 7 square array of rods and replace them by a

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sand glass shape of rods. The radius of rods forming the sand glass shape is R=0.154*a. A scattering dielectric-rod is placed in each corner of the formed cavity in order to suppress the counter propagating modes which can cause forged drops in the transmission spectrum. This

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cavity has been coupled with two input and output photonic waveguides.

3. Simulations and results

In the first part of simulations, we investigate the effect of nonlinear parameter (Kerr coefficient) over the proposed structure response . Therefore, , some centro-symetric rods were replaced by a non-centrosymetric material (GaAs) having the same refractive index of

n=3.59 [40]and a Kerr coefficient n2= 1.5 10-16 W/m2 [41](blue colored rods). In the first case (case1), we replaced the cavity outer rods GaAS. Otherwise, in the second case (case2), both cavity core and cavity outer rods were replaced by the non-centrosymetric material. For the first case, the injection of a high power resulted in the variation of the refractive index by a small step of ∆n=0.01. . The resonance wavelength is shifted from λ=1.5309 µm to

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λ=1.5331 µm, the transmission kept the same amplitude of about 96% and the Q-factor is

slightly increasing from 1913.25 in linear case to 1916.375 when ∆n=0.05. The interspaces

Case 1

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

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between resonant wavelengths are noticed to be 0.3 nm.

1.0

Delta n = 0 Delta n = 0.02

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0.8

Delta n = 0.03 Delta n = 0.04

A

0.7 0.6

Delta n = 0.05

0.5 0.4 0.3 0.2 0.1 0.0

1.528

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ED

M

Normalized Transmission (a.u.)

0.9

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Fig. 2 Nonlinear cavity based cavity

1.530

1.532

1.534

1.536

Wavelength (m)

Fig. 3 Normalized Transmission spectrum for different ∆n

The following table shows, the resonance wavelength and the Q-factor behaviors following

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the refractive index increase of nonlinear rods.

Table 1. Impact of refractive index change on the first proposed cavity characteristics ∆n

Resonance

∆λ (nm)

Q-factor

TM (%)

Wavelength

1.5318

0.8

1914.75

95.6

0.03

1.5322

0.8

1915.25

95.6

0.04

1.5326

0.8

1915.75

0.05

1.5331

0.8

1916.375

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0.02

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( µm)

96

N

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96

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Considering the second case, in which the refractive index of the cavity outer and core rods’

M

changed slightly, the resonant wavelength was shifted to 1.5339 µm when ∆n=0.05. A better increasing Q-factor was noticed reaching a value of 1917.375 when ∆n=0.05.

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Delta n=0

0.9

Normalized Transmission (a.u.)

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PT

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Case 2

Delta n=0.02

0.8 Delta n=0.03

0.7 Delta n=0.04

0.6 Delta n=0.05

0.5 0.4 0.3 0.2 0.1 0.0 1.528

1.530

1.532

1.534

1.536

Wavelength (m)

Fig. 5 Normalized Transmission spectrum for Fig. 4 Nonlinear cavity based switch Simulation results for the second case is resumed in the following table different ∆n Table 2. Impact of refractive index change on the second proposed cavity characteristics

∆n

Resonance

∆λ (nm)

Q-factor

TM (%)

Wavelength

1.5321

0.8

1915.125

95

0.03

1.5327

0.8

1915.875

95.06

0.04

1.5333

0.8

1916.625

95.2

0.05

1.5339

0.8

1917.375

95.31

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3.1 Switching process

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0.02

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( µm)

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In the second part of our simulations, in order to pick out the minimum ON/OFF switching

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threshold power ensuring the best performance, we conduct a study of the required power, the extinction ratio and the insertion loss behavior over different refractive index value changes.

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The following tables show the obtained results in both cavity structures.

Required Power T.M (%)

Extinction

Insertion

(W/µm2)

Ratio(dB)

Loss(dB)

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∆n

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Table 3. Switching results of the structure in Case1

0.02

A

0.03

0.04

130

200

266

ON

95.6

OFF

15

ON

95.6

OFF

10.6

ON

96

OFF

7

8.04 -13.56 9.55 -13.56

11.37

-13.98

0.05

333

ON

96

OFF

4.5

13.29

-13.98

Required Power T.M (%)

Extinction

Insertion

(W/µm2)

Ratio(dB)

Loss(dB)

333

11.5

ON

95.06

OFF

6.5

ON

95.2

OFF

4

ON

95

OFF

9.17

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OFF

-13.01

11.65

-13.06

13.77

-13.19

2.5

15.8

-13.19

ED

0.05

266

95.03

U

0.04

200

ON

N

0.03

130

A

0.02

M

∆n

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Table 4. Switching results of the structure in Case2

From the comparative switching study of the two cavity structures, it can be noticed that in

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the second case, the best extinction ratio and the best insertion loss are offered Moreover, the minimum switching threshold powers required for switching ON/OFF and ensures acceptable

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performances of transmission efficiency, extinction ratio and insertion loss in case 1 and case 2 are 200 W/µm2 and 130 W/µm2.

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In Figure 6 and Figure7, we show ON and OFF states at the operating wavelength λ= 1.5339µm in the second case. For off state, the low input power P =1W/µm2 is not sufficient to ensure the refractive index change in nonlinear rods. Contrary, =333W/µm2 engenders the resonant wavelength shift

a high input power P

toward λ= 1.5339µm . The

transmission spectrum reaches 95% and the switch is considered on its ON state.

(a) OFF State

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(b) ON State

N

1.0

0.6

0.2

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0.5

0.3

Nonlinear Case

M

0.7

0.4

Linear Case

A

0.8

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Normalized Transmission (a.u.)

0.9

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(b) Fig. 6 Light propagation in the proposed switch

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(a)

0.1

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0.0

1.526

1.528

1.530

1.532

1.534

1.536

1.538

Wavelength (m)

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Fig. 7 Output transmission spectra of proposed cavity based switch

3.2 Switch response time

As the switching time is defined by the time taken by the output signal to reach a steady level, Figure 8 presents the output signal time behavior at 1.5339µm at low and high injected power. When the power is low, the switch is on its OFF state and the switching time is Tsw1=8.33ps (cT=2500). Nevertheless, by injecting high power, the switch state

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M

A

N

U

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changes from OFF to ON with a switching time of Tsw2= 15ps (cT=4500)

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Fig. 8 ON/OFF time response of the proposed switch As a comparison, in the following table we resume the performances of the proposed switch

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and those previously published.

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Table 5. Comparison Table of the proposed Switch within recently reported results

Ref

T.M (%)

80 [36]

ON

Extinction

Insertion

Response

Ratio (dB)

Loss (dB)

Time (ps)

Q-factor

OFF 20

91

-6.98

26

-

-

-

1000-3000

10

-

1917.375

15.8

517

95

[39]

-

Our Work

ON

95.31

-13.19

15

A

N

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OFF 2.5

0.8-2.5

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[38]

6.02

IP T

[37]

2200

M

4. Conclusion

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In this paper, a novel hourglass nonlinear photonic crystal cavity structure is proposed. The designed cavity is demonstrated as an ultra-high all-optical switching process using

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FDTD method. Firstly, we studied the impact of nonlinear rods refractive index change on

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the transmission spectrum and the Q-factor in two photonic crystal cavities. Then, the work was continued to study the switching characteristics for different input power levels. A high injected power of 333 W/µm2 activated the Kerr effect and shifted the resonance

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wavelength toward high value resulting in the switching from OFF state to ON state. The obtained results show a short response time, a very low insertion loss and a very high extinction ratio of the proposed switch.

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