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Effective Index Method

Dr Mohamed Farhat

Professor, Senior Member IEEE, OSA and SPIE

Nanotechnology and Nanoelectronics Engineering Program

Zewail City of Science and Technology

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Slab and Channel Waveguides

n3 n1 n2

Planar (slab) waveguides for integrated photonics (e.g. laser chips)

(x, y, z)

(n1 > n2, n3)

x

z

y

y

x

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Effective Index Method (EIM)

  • The effective index method is an approximation technique for analyzing weakly guiding rectangular waveguides.
  • The EIM converts a single 2D problem into two 1D problems.

 

 

 

 

 

 

 

 

  • The method is a good approximation if the waveguide width is much larger than its thickness (W > 3d at least). So, this method is not applicable to square waveguides.

y

x

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Effective Index Method (EIM)

  • The planar structure defines TE and TM polarizations, but the lateral structure distorts them

 

  • When the aspect ratio (W > 3d) condition is satisfied, the characteristics of the guided modes are primarily determined by the thin slab (planar structure), much like a planar waveguide of thickness (d), but are modified by a lateral structure of width (W).

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Mathematical Analysis of the EIM

Starting form Maxwell’s wave equation:

Let:

Assuming that F(x) and G(y) are independent

Where:

y

x

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Mathematical Analysis of the EIM

 

 

 

 

 

 

 

 

 

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Mathematical Analysis of the EIM

The same as the slab waveguide problem

 

 

 

y

x

 

 

 

 

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Solution Strategy of the EIM

 

 

 

 

 

 

 

y

 

 

 

x

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Solution Strategy of the EIM

 

 

 

 

 

 

y

x

 

 

 

y

x

2- We have noticed that the structure has been rotated where the refractive index changes in x-direction.

 

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Solution Strategy of the EIM

 

 

 

 

y

x

 

 

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Solution Strategy of the EIM

One can follow the following procedure:

 

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Analysis of Rib Waveguide using EIM

 

 

 

 

W

1. Replace the 2D waveguide with a combination of 1D waveguide along x-axis.

 

 

 

 

 

 

 

y

x

 

 

 

 

 

 

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Analysis of Rib Waveguide using EIM

 

y

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Analysis of Rib Waveguide using EIM

4- Determine an effective index by solving the 1D waveguide constructed in step (3) along the x-axis.

 

 

 

y

x

W

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Analysis of Rib Waveguide using EIM

Firstly:

  • Divide the ridge waveguide to three vertical sections.
  • Solving for the Neff for the three sections using the conventional slab waveguide code for TE.

nSi= 3.5,

nSiGe = 3.50125.

λ = 1.32 μm.

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Analysis of Rib Waveguide using EIM

TE_Mode (12,6.5,12,3.5,3.50125,3.5,1.32,0)

​

Neff = 3.5008

TE_Mode (10,5.4,10,1,3.50125,3.5,1.32,0)

​

Neff = 3.4989

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Analysis of Rib Waveguide using EIM

nSi= 3.5,

nSiGe = 3.50125.

λ = 1.32 μm.

Secondly:

  • Replace the second section with the Neff resulted from step (1).
  • Solving for the Neff for the whole structure using the conventional slab waveguide code for TM.

 

3.5008

 

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Analysis of Rib Waveguide using EIM

TM_Mode (25,16,25, 3.4989,3.5008, 3.4989,1.32,0)

​

Neff = 3.5006

 

3.5008

 

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Plotting Total Field

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Ex.

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