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Mode Expansion
  • Mode Expansion

Mode Expansion

The section describes mode expansion.

The section includes basic principles of mode expansion and relevant settings in software's mode expansion.

Software supports the addition and modification of Mode expansion using scripts. Please refer to Script for more information.

Mode Expansion

Mode expansion allows for analyzing power ratio of any mode transmitted onto non-absorbing waveguide or fiber, and thus accurate results can be obtained from simulation.

Given a complete basic state of a mode, any input field can be represented as a linear combination of these modes.

Ein=∑m(afEmforward+abEmbackward)E_{in}=\sum_{m}(a_{f}E_{m}^{forward}+a_{b}E_{m}^{backward})

Hin=∑m(afHmforward+abHmbackward)H_{in}=\sum_{m}(a_{f} H_{m}^{forward}+a_{b}H_{m}^{backward})

Modes propagating forward and modes propagating backward are symmetric. Assuming mode propagation in zz direction, decompose EmforwardE_{m}^{forward} into a transverse component Et,mforwardE_{t,m}^{forward} and a longitudinal component Ez,mbackwardE_{z,m}^{backward}; and decompose HmforwardH_{m}^{forward} into a transverse component Ht,mforwardH_{t,m}^{forward} and a longitudinal component Hz,mbackwardH_{z,m}^{backward}.

Embackward=Et,mforward−Ez,mforwardz^E_{m}^{backward}=E_{t, m}^{forward}-E_{z, m}^{forward} \hat{z}

H∗mbackward=−H∗t,mforward+H_z,mforwardz^H*{m}^{backward}=-H*{t, m}^{forward}+H\_{z, m}^{forward} \hat{z}

Coefficient of a specific mode mm can be expressed by overlap integrals as:

0.5∗∫dS⋅Ein×Hm∗=(af+ab)Nm0.5*\int dS\cdot E_{in}\times H_{m}^{*}=(a_{f}+a_{b})N_{m}

0.5∗∫dS⋅Em∗×Hin=(af−ab)Nm∗0.5*\int dS \cdot E_{m}^{*}\times H_{in}=(a_{f}-a_{b})N_{m}^{*}

A user specifies arbitrary input fields EinE_{in} and HinH_{in}, as well as modal fields EmE_{m} and HmH_{m} (provided by mode extension monitor), and the following results may be returned by mode expansion:

af=0.25∗(∫dS⋅Ein×Hm∗Nm+∫dS⋅Em∗×HinNm∗)a_{f} = 0.25 * ( \frac{\int d S \cdot E _{in} \times H_{m} ^{*} }{N _{m}} + \frac{ \int dS \cdot E _{m} ^{*}\times H _{in}}{N _{m} ^{*}})

ab=0.25∗(∫dS⋅Ein×Hm∗Nm−∫dS⋅Em∗×HinNm∗)a_{b} = 0.25*(\frac{\int dS \cdot E_{in} \times H_{m}^{*}}{N_{m}} - \frac{\int dS \cdot E_{m} ^{*} \times H_{in}}{N _{m}^{*}})

Nm=0.5∗∫dS⋅Em×Hm∗N_{m} = 0.5*\int dS \cdot E_{m}\times H _{m}^{*}

Pin=0.5∗∫dS⋅Ein×Hin∗P_{in} = 0.5*\int dS \cdot E_{in}\times H_{in}^{*}

afa_f and aba_b represent complex transmission coefficients for forward and backward propagating light waves, respectively. NmN_{m} is power of mode mm, and PinP_{in} is total input power.

Mode Expansion Setting

Rotation

Name Description
Angle theta Angle between propagating direction and incident plane normal.
Angle phi Angle at which propagating direction rotates clockwise around incident plane normal.
Offset Set plane offset to calculate mode.

Select Mode

By default, the FDFP monitor or Port selects the Fundamental mode (the mode with the largest effective index). You can also control the polarization of the automatically selected mode by choosing Fundamental TE Mode or Fundamental TM Mode in Mode Selection in the Mode expansion tab of the FDFP monitor or the General tab of the Port. The mode selected using these settings will automatically update when the structure is modified. You can click Update Modes to view the changes. This is particularly useful for tasks such as parameter sweeps, where the waveguide geometry is changed.

Select "user select" in Mode Selection allows you to choose any mode. In this case, the mode solver can be launched by clicking the Select mode button in the Mode expansion tab of the FDFP monitor or the General tab of the Port.

Mode List

Mode list tab displays mode information resulting from solving modes.

Name Description
Mode# Serial number of mode.
Effective index Effective index of mode.
TE/TM fraction Relative power distribution between TE and TM modes.
Loss Mode transmission loss, which refers to power attenuation of optical signals in waveguide devices due to absorption, scattering, and other factors.
Wavelength Wavelength corresponding to a mode.
Mode name Mode name.

Beam Profile

Beam profile tab displays an image based on resulting data of the selected mode.

Name Description
Results Allow users to specify type of data to plot by drop-down selection of either Mode fields or Index.
Components When selecting Mode fields for Results, available results are E Magnitude, Ex, Ey, Ez, H Magnitude, Hx, Hy, Hz; When selecting Index for Results, available results are Index_x, Index_y, Index_z.
Scalar Abs: modulus of the selected component; Re: real part of the selected component; Im: imaginary part of the selected component; Phase: argument of the selected component.
Refresh Refresh profile.
Visualize data Open Data Visualize window.
Plot in new window Plot profile in a new window.

General

General tab contains the following settings:

Name Description
Incident axis Incident axis of mode source (normal of incident plane), a read-only parameter.
Total modes Set number of modes to be solved.
  • Frequency
Name Description
Frequency Frequency of mode solving, a read-only parameter.
Wavelength Wavelength of mode solving, a read-only parameter.
  • Hypothesis
Name Description
Use max refractive index Use maximum refractive index of structure for mode calculation.
Guess value Specify refractive index for mode calculation; used to compute a mode close to a user-defined value of effective index; enabled when Use max refractive index is unchecked.
  • Bent waveguide

Enabled when mode of bent waveguide is solved.

Name Description
Bend radius Radius of curvature of bent waveguide.
Bend waveguide width Width of bent waveguide.
  • TE/TM
Name Description
Quasi-TE Quasi-transverse electric field mode; Electric field of Quasi-TE mode is mainly concentrated in the direction (transverse) perpendicular to propagation direction, with possible existence of some non-zero electric field components along propagation direction (longitudinal). On the other hand, electric field of TE mode is completely concentrated in the direction (transverse) perpendicular to propagation direction, with no electric field components along propagation direction (longitudinal).
Quasi-TM Quasi-transverse magnetic field mode; Magnetic field of Quasi-TM mode is mainly concentrated in the direction (transverse) perpendicular to propagation direction, with possible existence of some non-zero magnetic field components along propagation direction (longitudinal). On the other hand, magnetic field of TM mode is completely concentrated in the direction (transverse) perpendicular to propagation direction, with no magnetic field components along propagation direction (longitudinal).
All All modes.
  • Solve modes
Name Description
Solve modes By clicking on this button, mode solver solves modes based on parameters set by a user.

Boundary

Enabled when Override default boundary tab is checked.

Name Description
X/Y axis min Select type of boundary condition: PML, Periodic, PEC, PMC.
X/Y axis max Select type of boundary condition: PML, Periodic, PEC, PMC.

For more information about boundary conditions, please refer to Boundary Conditions.

Advanced

Advanced tab allows you to set different sampling methods and automatically remove PML mode.

Name Description
Sampled material sampling type Select sampling method of material data by drop-down: Linear interpolation: Obtain material data using linear interpolation. Obtained material data may be discontinuous in frequency; Fit model: Fit Sampled material data using a multi-coefficient material model.
Analytic material sampling type Select sampling method of material data by drop-down: Analytic model: use a model data of Analytic material; Fit model: use multi-coefficient material model to fit Analytic material data.
Automatically remove PML modes Automatically remove PML mode.
Threshold for PML mode removal A threshold required to remove PML mode.