3.1. fmas.models

Implements several \(z\)-propagation models based on the forward model for the analytic signal [1,2,3,4].

ModelBaseClass	Base class for propagation models.
FMAS_THG	Forward model for analytic signal including third-harmonic generation.
FMAS	Forward-model for the analytic signal.
FMAS_S	Simplified variant of the forward-model for the analytic signal.
FMAS_S_Raman	Simplified variant of the forward-model for the analytic signal including the Raman effect.
BMCF	Bidirectional model for the complex field.
CustomModelPCF	Custom model for specific Photonic Crystal Fiber

Further \(z\)-propagation models can be implemented by using the class ModelBaseClass.

References

[1] Sh. Amiranashvili, A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses, Phys. Rev. E 10 (2010) 013812, http://dx.doi.org/10.1103/PhysRevA.82.013812.

[2] Sh. Amiranashvili, A. Demircan, Ultrashort Optical Pulse Propagation in terms of Analytic Signal, Adv. Opt. Tech. 2011 (2011) 989515, http://dx.doi.org/10.1155/2011/989515.

[3] A. Demircan, Sh. Amiranashvili, C. Bree, C. Mahnke, F. Mitschke, G. Steinmeyer, Rogue wave formation by accelerated solitons at an optical event horizon, Appl. Phys. B 115 (2014) 343, http://dx.doi.org/10.1007/s00340-013-5609-9

[4] A. Demircan, Sh. Amiranashvili, C. Bree, U. Morgner, G. Steinmeyer, Supercontinuum generation by multiple scatterings at a group velocity horizon, Opt. Exp. 22 (2014) 3866, https://doi.org/10.1364/OE.22.003866.

class fmas.models.BMCF(w, beta_w, chi=1.0)

Bases: fmas.models.model_base.ModelBaseClass

Bidirectional model for the complex field.

Implements the bidirectional model for the complex field (BMCF), i.e. Eq. (31) of Ref.[1]. It includes third-harmonic generation and self-steepening for interacting forward and backward components of the optical field.

References

[1] Sh. Amiranashvili, A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses, Phys. Rev. E 10 (2010) 013812, http://dx.doi.org/10.1103/PhysRevA.82.013812.

Parameters

• w (numpy.ndarray) – Angular frequency grid.

• beta_w (numpy.ndarray) – Propagation constant.

• chi (float) – Nonlinear susceptibility (default=1.0).

property Lw

Frequency-domain representation of nonlinear operator.

Returns

Frequency-domain representation of linear operator of the partial differential equation.

Return type

numpy.ndarray

Nw(uw)

Frequency-domain representation of nonlinear operator.

Parameters

uw (numpy.ndarray) – Frequency-domain representation of field at current \(z\)-position.

Returns

Frequency-domain representation of field at current \(z\)-position.

Return type

numpy.ndarray

claw(i, zi, w, uw)

Conservation law of the propagation model.

Implements conserved quantity related to the field energy, given by

\[C_{\mathcal{E}}(z) = \sum_\omega |\mathcal{E}_\omega(z)|^2.\]

Parameters

• i (int) – Index specifying the current \(z\)-step.

• zi (float) – Current \(z\)-value.

• w (numpy.ndarray) – Angular frequency mesh.

• uw (numpy.ndarray) – Freuqency domain representation of the current field.

Returns

value of the conserved quantitiy.

Return type

numpy.ndarray

class fmas.models.CustomModelPCF(w)

Bases: fmas.models.fmas_s_raman.FMAS_S_Raman

Custom model for specific Photonic Crystal Fiber

class fmas.models.FMAS(w, beta_w, chi=1.0, alpha_w=0.0)

Bases: fmas.models.model_base.ModelBaseClass

Forward-model for the analytic signal.

Implements the forward-model for the analytic signal (FMAS) [1,2]. In particular, this model implements Eq. (33) of Ref.[1].

References

[1] Sh. Amiranashvili, A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses, Phys. Rev. E 10 (2010) 013812, http://dx.doi.org/10.1103/PhysRevA.82.013812.

[2] Sh. Amiranashvili, A. Demircan, Ultrashort Optical Pulse Propagation in terms of Analytic Signal, Adv. Opt. Tech. 2011 (2011) 989515, http://dx.doi.org/10.1155/2011/989515.

Parameters

• w (numpy.ndarray) – Angular frequency grid.

• beta_w (numpy.ndarray) – Propagation constant.

• alpha_w (numpy.ndarray) – Frequency-domain representation of root-power loss.

• chi (float) – Nonlinear susceptibility (default=1.0).

property Lw

Frequency-domain representation of nonlinear operator.

Returns

Frequency-domain representation of linear operator of the partial differential equation.

Return type

numpy.ndarray

Nw(uw)

Frequency-domain representation of nonlinear operator.

Parameters

uw (numpy.ndarray) – Frequency-domain representation of field at current \(z\)-position.

Returns

Frequency-domain representation of field at current \(z\)-position.

Return type

numpy.ndarray

claw(i, zi, w, uw)

Conservation law of the propagation model.

Implements conserved quantity related to the field energy, given by

\[C_{\mathcal{E}}(z) = \sum_\omega |\mathcal{E}_\omega(z)|^2.\]

Parameters

• i (int) – Index specifying the current \(z\)-step.

• zi (float) – Current \(z\)-value.

• w (numpy.ndarray) – Angular frequency mesh.

• uw (numpy.ndarray) – Freuqency domain representation of the current field.

Returns

value of the conserved quantitiy.

Return type

numpy.ndarray

class fmas.models.FMAS_S(w, beta_w, n2=1.0, alpha_w=0.0)

Bases: fmas.models.model_base.ModelBaseClass

Simplified variant of the forward-model for the analytic signal.

Implements a simplified variant of the forward-model for the analytic signal [1,2]. In particular, this model implements Eq. (9) of Ref.[2].

References

[1] Sh. Amiranashvili, A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses, Phys. Rev. E 10 (2010) 013812, http://dx.doi.org/10.1103/PhysRevA.82.013812.

[2] A. Demircan, Sh. Amiranashvili, C. Bree, C. Mahnke, F. Mitschke, G. Steinmeyer, Rogue wave formation by accelerated solitons at an optical event horizon, Appl. Phys. B 115 (2014) 343, http://dx.doi.org/10.1007/s00340-013-5609-9

Parameters

• w (numpy.ndarray) – Angular frequency grid.

• beta_w (numpy.ndarray) – Propagation constant.

• alpha_w (numpy.ndarray) – Frequency-domain representation of root-power loss.

• n2 (float) – Nonlinear refractive index (default=1.0).

property Lw

Frequency-domain representation of nonlinear operator.

Returns

Frequency-domain representation of linear operator of the partial differential equation.

Return type

numpy.ndarray

Nw(uw)

Frequency-domain representation of nonlinear operator.

Parameters

uw (numpy.ndarray) – Frequency-domain representation of field at current \(z\)-position.

Returns

Frequency-domain representation of field at current \(z\)-position.

Return type

numpy.ndarray

claw(i, zi, w, uw)

Conservation law of the propagation model.

Implements conserved quantity related to the field energy, given by

\[C_{\mathcal{E}}(z) = \sum_\omega |\mathcal{E}_\omega(z)|^2.\]

Parameters

• i (int) – Index specifying the current \(z\)-step.

• zi (float) – Current \(z\)-value.

• w (numpy.ndarray) – Angular frequency mesh.

• uw (numpy.ndarray) – Freuqency domain representation of the current field.

Returns

value of the conserved quantitiy.

Return type

numpy.ndarray

fmas.models.FMAS_S_R

alias of fmas.models.fmas_s_raman.FMAS_S_Raman

class fmas.models.FMAS_S_Raman(w, beta_w, n2=1.0, fR=0.18, tau1=12.2, tau2=32.0, alpha_w=0.0)

Bases: fmas.models.model_base.ModelBaseClass

Simplified variant of the forward-model for the analytic signal including the Raman effect.

Implements a simplified variant of the forward-model for the analytic signal including the Raman effect [1]. In particular, this model implements Eq. (10) of Ref.[1].

Aliased as FMAS_S_R.

References

[1] A. Demircan, Sh. Amiranashvili, C. Bree, C. Mahnke, F. Mitschke, G. Steinmeyer, Rogue wave formation by accelerated solitons at an optical event horizon, Appl. Phys. B 115 (2014) 343, http://dx.doi.org/10.1007/s00340-013-5609-9

Parameters

• w (numpy.ndarray) – Angular frequency grid.

• beta_w (numpy.ndarray) – Propagation constant.

• alpha_w (numpy.ndarray) – Frequency-domain representation of root-power loss.

• n2 (float) – Nonlinear refractive index (default=1.0).

• fR (float) – Fractional raman response (default=0.18).

• tau1 (float) – Time scale associated with oscillator angular frequency in Lorentz model of Raman response (default=12.2 fs).

• tau2 (float) – Time scale associated with oscillator angular frequency in Lorentz model of Raman response (default=32.0 fs).

property Lw

Frequency-domain representation of nonlinear operator.

Returns

Frequency-domain representation of linear operator of the partial differential equation.

Return type

numpy.ndarray

Nw(uw)

Frequency-domain representation of nonlinear operator.

Parameters

uw (numpy.ndarray) – Frequency-domain representation of field at current \(z\)-position.

Returns

Frequency-domain representation of field at current \(z\)-position.

Return type

numpy.ndarray

claw(i, zi, w, uw)

Conservation law of the propagation model.

Implements conserved quantity related to the field mass, given by

\[C_{\mathcal{E}}(z)=\sum_{\omega>0}|\mathcal{E}_\omega(z)|^2/\omega.\]

Parameters

• i (int) – Index specifying the current \(z\)-step.

• zi (float) – Current \(z\)-value.

• w (numpy.ndarray) – Angular frequency mesh.

• uw (numpy.ndarray) – Freuqency domain representation of the current field.

Returns

value of the conserved quantitiy.

Return type

numpy.ndarray

class fmas.models.FMAS_THG(w, beta_w, chi=1.0, alpha_w=0.0)

Bases: fmas.models.model_base.ModelBaseClass

Forward model for analytic signal including third-harmonic generation.

Implements the forward model for the analytic signal including third-harmonic generation, see Eq. (17) of Ref.[1].

References

[1] A. Demircan, Sh. Amiranashvili, C. Bree, U. Morgner, G. Steinmeyer, Supercontinuum generation by multiple scatterings at a group velocity horizon, Opt. Exp. 22 (2014) 3866, https://doi.org/10.1364/OE.22.003866.

Parameters

• w (numpy.ndarray) – Angular frequency grid.

• beta_w (numpy.ndarray) – Propagation constant.

• chi (float) – Nonlinear susceptibility (default=1.0).

property Lw

Frequency-domain representation of nonlinear operator.

Returns

Frequency-domain representation of linear operator of the partial differential equation.

Return type

numpy.ndarray

Nw(uw)

Frequency-domain representation of nonlinear operator.

Parameters

uw (numpy.ndarray) – Frequency-domain representation of field at current \(z\)-position.

Returns

Frequency-domain representation of field at current \(z\)-position.

Return type

numpy.ndarray

claw(i, zi, w, uw)

Conservation law of the propagation model.

Implements conserved quantity related to the field energy, given by

\[C_{\mathcal{E}}(z) = \sum_\omega |\mathcal{E}_\omega(z)|^2.\]

Parameters

• i (int) – Index specifying the current \(z\)-step.

• zi (float) – Current \(z\)-value.

• w (numpy.ndarray) – Angular frequency mesh.

• uw (numpy.ndarray) – Freuqency domain representation of the current field.

Returns

value of the conserved quantitiy.

Return type

numpy.ndarray

class fmas.models.ModelBaseClass(w, beta_w, alpha_w=None)

Bases: object

Base class for propagation models.

w

Angular frequency grid.

Type

numpy.ndarray

beta_w

Frequency-domain representation of propagation constant.

Type

numpy.ndarray

alpha_w

Frequency-domain representation of root-power loss.

Type

numpy.ndarray

c0

speed of light

Type

float

Parameters

• w (numpy.ndarray) – Angular frequency grid.

• beta_w (numpy.ndarray) – Frequency-domain representation of propagation constant.

• alpha_w (numpy.ndarray) – Frequency-domain representation of root-power loss (default: None).

property Lw

Frequency-domain representation of nonlinear operator.

Returns

exception NotImplementedError

Nw(uw)

Frequency-domain representation of nonlinear operator.

Parameters

uw (numpy.ndarray) – Frequency-domain representation of field at current \(z\)-position.

Returns

exception NotImplementedError

claw(*args)

Conservation law.

Callback function that can be used to implementing a measurement using a user-supplied function.

Returns

None