Slot Waveguide

Research in the field of photonic integrated circuits (PICs) is taking a boost, especially because of its compatibility with the existing CMOS fabrication techniques using materials such as silicon (Si) and silicon dioxide (SiO2). Doing extensive numerical simulations on the photonic component design before fabricating the final prototype essentially saves resources.

Optical waveguides are extensively used in PICs. They have the responsibility to transfer optical energy and signals from one optical component to another. Exhaustive research has been performed with a particular type of configuration of the optical waveguide where the high refractive index medium (core) is wrapped around by a low refractive index medium (cladding). The physics behind transferring the energy in such core/cladding configuration is simply based on total internal reflection (TIR). For more information about traditional optical waveguides, see for example Ref. 1.

However counterintuitive, research is also carried out where the optical energy is made to confine within the low refractive index slot placed bordering two high refractive index slabs as shown in the Figure 1. This is the slot waveguide configuration.

Figure 1: The slot waveguide geometry, also indicating the materials and their respective refractive indices.

Maxwell’s equations state that the normal component of the electric displacement field (the D field) must be continuous across interfaces between materials with different refractive indices. Thus, the normal component of the electric field (E field) must be higher in the material with the low refractive index. This can be used to enhance and confine the guided mode into a narrow domain with low refractive index — the slot domain.

The dimension of this low refractive index domain is in orders of tens of nanometers which keeps the optical energy tightly confined to a narrow area giving a high optical energy density in the waveguide.

For more information about this slot waveguide structure, see Ref. 2.

Model Definition

Mode analysis is performed on the cross section of the optical waveguide rather than modeling the complete 3D geometry.

The wave is propagating out of the plane in z direction, as shown below

(1)

where ω is the angular frequency and β is the propagation constant.

The eigenvalue equation for the electric field is obtained from the Helmholtz equation

(2)

The above equation is solved for the eigenvalue λ = -jβ.

As shown in Figure 1, the silicon dioxide slot has a refractive index of 1.44, while the neighboring silicon slabs has a refractive index of 3.48.

After the mode analysis was performed, to optimize the width of the nano-slot to provide the maximum optical power through the waveguide, a parametric sweep of the width from 30 nm to 140 nm was performed.

To evaluate the normalized optical power and optical intensity through the waveguide, two integration operators were defined; first to perform an integration over the slot area and second for the complete waveguide area. A maximum operator was used to evaluate the normalized transverse electric field (Ex) through the center of the waveguide.

Results and Discussion

The mode analysis evaluates the fundamental mode for a slot width of 50 nm at an operating wavelength of 1.55 μm. The surface plot showcases the in-plane transverse electric field (Ex) confined in the narrow slot as shown in the Figure 2.

Figure 2: Schematic of the 50 nm width slot waveguide configuration with the transverse electric field (Ex) surface plot. The effective mode index is 1.8247.

The ratio of Ex over the absolute maximum of Ex was used to evaluate the normalized x component of the transverse electric field through the center of the waveguide, as shown in Figure 3. A large discontinuity in the electric field can be observed specifically at ±25 nm.

Figure 3: Normalized transverse electric field (Ex) through the center of the waveguide.

To visualize this discontinuity more comprehensively, a surface plot along with the height expression was plotted as shown in Figure 4.

Figure 4: Surface plot along with its height expression to visualize a 3D representation of the transverse electric field Ex.

Finally, the normalized power and intensity through the waveguide with respect to the different slot width is highlighted in Figure 5. The normalized quantities were derived as the ratio of integrated optical power and optical intensity in the slot over the integrated optical power and optical intensity through the complete waveguide. It could be emphasized that the normalized optical power peaks for the slot width between 50 nm and 120 nm.

Figure 5: Normalized optical power and optical intensity through the slot with respect to the slot width for an operating wavelength of 1.55 μm.

References

1. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, John Wiley & Sons, Inc., chap. 7, 1991.

2. V. Almeida, Q. Xu, C. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure”, Optics Letters, vol. 29, pp. 1209–1211, 2004.

Wave_Optics_Module/Waveguides_and_Couplers/slot_waveguide

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

Start by adding some parameters that will simplify the setup of the geometry, the materials, and the study.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
w_slab	180[nm]	1.8E-7 m	Width of slab
w_slot	50[nm]	5E-8 m	Width of slot
h_slot	300[nm]	3E-7 m	Height of slot
lda0	1.55[um]	1.55E-6 m	Operating wavelength
f0	c_const/lda0	1.9341E14 1/s	Operating frequency
n_slab	3.48	3.48	Refractive index of slab
n_slot	1.44	1.44	Refractive index of slot
n_clad	1.44	1.44	Refractive index of cladding
a	2[um]	2E-6 m	Exterior domain side length
b	800[nm]	8E-7 m	Interior domain width
c	500[nm]	5E-7 m	Interior domain height

Geometry 1

The geometry consists of a number of rectangles defining the slot, the slabs, and the cladding domain.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type w_slot.

In the text field, type h_slot.

Locate the section. From the list, choose .

Click

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type w_slab.

In the text field, type h_slot.

Locate the section. In the text field, type w_slot/2.

In the text field, type -h_slot/2.

Click

Move 1 (mov1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Select the check box.

Locate the section. In the text field, type -w_slot-w_slab.

Click

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type a.

Locate the section. From the list, choose .

Click

Click the

button in the toolbar.

Rectangle 4 (r4)

In the toolbar, click

In the window for , locate the section.

In the text field, type b.

In the text field, type c.

Locate the section. From the list, choose .

Click

Line Segment 1 (ls1)

Add a line segment to define a line through the center of the waveguide.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type -b/2.

Locate the section. From the list, choose .

In the text field, type b/2.

Click

Definitions

Now, add an operator taking the maximum value of a variable defined on the centerline through the waveguide.

Maximum 1 (Center)

In the toolbar, click

and choose .

In the window for , type Maximum 1 (Center) in the text field.

Locate the section. From the list, choose .

Select Boundaries 7, 12, 17, 22, and 26 only.

Integration 1 (Slot)

In the toolbar, click

and choose .

This integration operator should be defined for the slot domain.

In the window for , type Integration 1 (Slot) in the text field.

Select Domains 6 and 7 only.

Integration 2 (Complete Waveguide)

In the toolbar, click

and choose .

This integration operator is defined for the whole waveguide structure.

Click in the window and then press Ctrl+A to select all domains.

In the window for , type Integration 2 (Complete Waveguide) in the text field.

Materials

Now, define the materials in the waveguide structure.

Slot

In the window, under right-click and choose .

In the window for , type Slot in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_slot	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Slab

Right-click and choose .

In the window for , type Slab in the text field.

Select Domains 4, 5, 8, and 9 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_slab	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Cladding

Right-click and choose .

In the window for , type Cladding in the text field.

Select Domains 1–3 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_clad	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Mesh 1

Define a manual mesh sequence. The physics-controlled mesh sequence will be the starting point.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Size

In the window, under click .

In the window for , locate the section.

Click the button.

Size 1

This size node will set the maximum mesh element size for the center part of the waveguide structure to be one twentieth of its height.

In the window, click .

In the window for , locate the section.

From the list, choose .

Select Domains 2–9 only.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type c/20.

Click

Study 1

Step 1: Mode Analysis

In the window, under click .

In the window for , locate the section.

In the text field, type f0.

Select the check box.

In the associated text field, type 2.

Select the check box.

In the associated text field, type n_slab.

Add a parameteric sweep over the slot width.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
w_slot (Width of slot)		m

Click to select row number 1 in the table.

Click

In the dialog box, type 30[nm] in the text field.

In the text field, type 10[nm].

In the text field, type 140[nm].

Click .

In the window for , locate the section.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
w_slot (Width of slot)	range(30[nm],10[nm],140[nm])	nm

In the toolbar, click

Results

Electric Field (ewfd)

In the window for , locate the section.

From the list, choose .

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type ewfd.Ex.

Contour 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type ewfd.Ex.

Locate the section. From the list, choose .

From the list, choose .

Clear the check box.

Arrow Surface 1

Right-click and choose .

Selection 1

In the window, right-click and choose .

Select Domains 6 and 7 only.

Arrow Surface 1

In the window, click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Find the subsection. In the text field, type 2.

Find the subsection. In the text field, type 30.

Locate the section. Select the check box.

In the associated text field, type 50000.

From the list, choose .

Electric Field (ewfd)

In the window, click .

In the window for , click to expand the section.

From the list, choose .

In the text area, type Surface and Countour: E<sub>x</sub> (V/m). Arrow Surface: Electric field. Slot width: 50 nm.

Click the

button in the toolbar twice to see the electric field enhancement in the slot region and to verify that the main electric field component indeed is the x component.

In the toolbar, click

Add a Cut Line dataset for later use in a line graph.

Cut Line 2D 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In row , set to -b/2.

In row , set to b/2.

Click

Normalized Transverse Electric Field

In the toolbar, click

In the window for , type Normalized Transverse Electric Field in the text field.

Locate the section. From the list, choose . Using the Cut Line dataset instead of a line selection will make the line appear as continuous also at points where the data actually is discontinuous.