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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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