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0),()(),( 22

2

2

=∂∂−∆ trEtc

rntrE

Wave equation in scalar approximation:

Beam Propagation Method

Scalar Helmholtz equation:

Propagation mainly in z-direction fast oscillations in z-direction SVE ansatz

Integrable equation:

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Example: Plane Wave

Beam Propagation Method

Scalar Helmholtz equation:

Propagation mainly in z-direction fast oscillations in z-direction SVE ansatz

Integrable equation:

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Beam Propagation Method

Restrictions of the basic method

-Paraxiality condition

-Scalar Waves

-No reflection

Lifted by

-Wide-Angle BPM (Padé approximation)

-Vector BPM

-Bidirectional BPM

Mode Solving with BPM

z‘ = -iz

Exponential growth, domination of lowest order mode at sufficient z

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Task 1• Simulate the light propagation in a system of two coupled waveguides.• Demonstrate the periodicity of the dynamics. (For excitations of the fundamental

modes in the guides.)Parameters: rectangular waveguide 5µm x 5µm; background index 1.5; index difference 0.01; wavelength 1.5 µm; distance 10 µm

Task 2• Add one more waveguide and simulate the light propagation in a system of three

coupled waveguides.• Show that depending on the symmetry of the excitation the dynamics can be

periodic or quasi-periodic.

Task 3• Add more guides to simulate the light propagation in a system of 101 waveguides.

RSoft: Waveguide Systems

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RSoft: Computing Modes in a Waveguide with BPM

1. Place the waveguide. Define its dimensions and refractive index.

2. Place launch field shifted to waveguide in order to be able to excite all modes. Use a Gaussian launch field.

3. Set wavelength (global variable).

4. Compute modes. Select in ‘Mode options‘ which modes are to be computed.

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RSoft: Computing Modes in a Waveguide with BPMData browser to access simulation results(select filename first!)

Using a wavelength of 500 nm there areseveral modes.

Usually they should be found in the order oftheir propagation constant (neff).

At a wavelength of 1500 nm the waveguide issingle mode!

Numerical convergence problems

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RSoft: Computing Modes in a Waveguide with BPMFor two coupled singlemode waveguides there exist two modes of the waveguide system:

Symmetric Mode Antisymmetric Mode

A launch field can be decomposed in guided modes and non-guided (leaky, evanescent) waves. Exciting one waveguide is mainly a superposition of the two system modes (symmetric and antisymmetric), both with about the same intensity. The excitation strength of each mode does not change during propagation. However, due to their different propagation constants βm the field distribution as a superposition of the modes

does change. The field intensity thus oscillates between the two waveguides.

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β

β0+2c

β0-2c

β0

Mode picture

Three Coupled Waveguides

Propagation constant ofsingle waveguide

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Periodicity vs. Quasi-Periodicity

Two waveguideszizi excexczx 21 )()(),( 2211

ββ φφφ +=

( )zizi excxcezx )(2211 121 )()(),( βββ φφφ −+=nz πββ 2)( 12 =−

Period12

2ββ

π−

=z

Three waveguideszizizi excexcexczx 321 )()()(),( 332211

βββ φφφφ ++=

( )zizizi excexcxcezx )(33)(2211 13121 )()()(),( βββββ φφφφ −− ++=

mznz

πββπββ2)(2)(

13

12

=−=−

nm=

−−

13

12

ββββ

Condition on z:

Demand for rational ratio of propagation constants

Not given in general (real values) Quasi-Periodicity

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Task 4• Simulate the Green's-Function of the system.

Task 5• Study the dispersion relation of the system.• Use a Gaussian beam excitation for which you vary the incidence angle at the input

and monitor the resulting output.• Search for angles of diffractionless propagation. What is the phase pattern of the

beam in this case.

RSoft: Waveguide Array

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Given

Propagation: Homogeneous Medium vs. Waveguide Array

Scalar Helmholtz equation

)0(nu

[ ] 0)()()( 11 =++ −+ zuzuczudzdi nnn

∫−

=π

π

κ κκ dezuzu nin );(~)(

( ) 0);(~ =

++ − zueecdzdi n

ii κκκ

κγ cos2c=

Solution zinn euzu)()0;(~);(~ κγκκ =

)0,,( yxu

0),,()(),,( 2 =+∆ zyxukzyxu ω

∫ ∫ += βαβα βα ddezUzyxu yxi )();,(),,(

0);,()(222

=

+ zU

dzd βαωγ

222 )( βαωγ −−= k

zieUzU ),(0 ),();,(βαγβαβα =

Mode excitation strength in singlemode waveguides (discrete numbers) at z=0

Field at z=0

Coupled modes equationPropagationequation

Fourier decomposition(plane waves)

Fourier transform in (x, y)

Fourier ansatzin prop. eqn.

Computationsequence

Fourier expansion in n

),,();,(),()0,,( 0zyxuzU

Uyxu→

→→βα

βα )();(~)0;(~)0( zuzuuu nnnn →→→ κκRefraction & DiffractionRefraction & Diffraction

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Propagation: Homogeneous Medium vs. Waveguide Array

κγ cos2c=222 )( βαωγ −−= k

RefractionRefraction & Diffraction & Diffraction κβ cos20 ckz +=(β0 … Prop. constant in uncoupled waveguide)

α

β

γ

-1 0 1ktd [π ]

k z

coupled systemsingle waveguide

κ=

Excitation of only one κ value (all waveguides with same excitation strength and same phase)

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-1 0 1ktd [π ]

k z

coupled systemsingle waveguide

Propagation: Homogeneous Medium vs. Waveguide Array

κγ cos2c=222 )( βαωγ −−= k

κβ cos20 ckz +=(β0 … Prop. constant in uncoupled waveguide)

α

β

γ

κ=

Excitation of only one κ value (all waveguides with same excitation strength but a phase shift)

RefractionRefraction & Diffraction & Diffraction

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trans

vers

al c

oord

inat

e

-1 0 1ktd [π ]

k z

coupled systemsingle waveguide

Propagation: Homogeneous Medium vs. Waveguide Array

κγ cos2c=222 )( βαωγ −−= k

κβ cos20 ckz +=(β0 … Prop. constant in uncoupled waveguide)

α

β

γ

κ=

Excitation of all κ values by excitation of only one waveguide (discrete delta peak)

RefractionRefraction & Diffraction & Diffraction

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-1 0 1ktd [π ]

k z

coupled systemsingle waveguide

Propagation: Homogeneous Medium vs. Waveguide Array

κγ cos2c=222 )( βαωγ −−= k

κβ cos20 ckz +=(β0 … Prop. constant in uncoupled waveguide)

κ=

Diffraction proportional to second derivative of diffraction relation

(a)

(b)

(c)

Refraction & Refraction & DiffractionDiffraction

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