Calculation of effective area for Spontaneous Four-wave Mixing (SFWM) #130

Kunyuan-LI · 2024-02-23T13:07:15Z

Hello,

Just as @elizaleung830 I'm trying to calculate the effective area, but for Spontaneous Four-wave Mixing (SFWM), which is a third-order non-linear effect and the Aeff needs to take into account the interaction between the electric field distributions of three modes (we consider a degenerate pump, which means there are two identical pumps).

The goal is to replicate the results found in the Ansys Lumerical tutorial on Spontaneous Four-Wave Mixing (SFWM) in a Microring Resonator Photon Source, as documented at https://optics.ansys.com/hc/en-us/articles/15100783091731, following the methodology outlined in the publication https://doi.org/10.1070/QEL16511:

$$A_{\mathrm{eff}}=\frac{1}{\iint \mathrm{d} x \mathrm{~d} y u_p(x, y) u_p(x, y) u_s^*(x, y) u_i^*(x, y)}$$

Where $u_p$, $u_s$, $u_i$ are the mode function describing the transverse spatial distribution of the field and normalised $\int|u(x, y)|^{2} \mathrm{~d} x \mathrm{~d} y=1$.

So, for a mode in femwell, does mode.E correspond to the unnormalized electric field distribution? I added normalization in femwell.maxwell.waveguide and calculated the method for Aeff, but I obtained a rather strange result.

Function added (for testing I unfreeze the class to add a E0 for the normalized electric field):

def calculate_sfwm_Aeff(
    basis: Basis,
    mode_p,
    mode_s,
    mode_i,
) -> np.complex64:
    """
    Calculates the overlap integral for SFWM process by considering the interactions
    between pump, signal, and idler modes in the xy plane.

    Args:
        basis (Basis): Common basis used for all modes in the same geometric structure.
        mode_p, mode_s, mode_i: Mode instances for pump, signal, and idler, respectively.

    Returns:
        np.complex64: The overlap integral result for the SFWM process.
    """
    def normalize_mode(mode):
        @Functional
        def E2(w):
            return (w["E"][0] * np.conj(w["E"][0]))


        E_xy, _ = mode.basis.interpolate(mode.E) #xy, z
        E_squared_integral = E2.assemble(mode.basis, E=E_xy)
        normalization_factor = 1 / np.sqrt(E_squared_integral)
        mode.E0 = mode.E * normalization_factor
    
    normalize_mode(mode_p)
    normalize_mode(mode_s)
    normalize_mode(mode_i)

    #Normalization needed for uv(x,y)!
    E_p, _ = mode_p.basis.interpolate(mode_p.E0) #xy, z
    E_s, _ = mode_s.basis.interpolate(mode_s.E0)
    E_i, _ = mode_i.basis.interpolate(mode_i.E0)
    
    @Functional(dtype=np.complex64)
    def sfwm_overlap(w):
        overlap_SFWM = w["E_p"][0] * w["E_p"][0] * np.conj(w["E_s"][0]) * np.conj(w["E_i"][0])
        return (overlap_SFWM)

    # Assemble the integral over the basis to compute the overlap
    overlap_result = sfwm_overlap.assemble(
        basis,
        E_p=E_p,
        E_s=E_s,
        E_i=E_i,
    )

    return  1/overlap_result

And for the main code:

from collections import OrderedDict
import matplotlib.pyplot as plt
import numpy as np
from scipy.constants import speed_of_light
from shapely.geometry import box
from shapely.ops import clip_by_rect
from skfem import Basis, ElementTriP0
from skfem.io.meshio import from_meshio
from tqdm import tqdm
from femwell.maxwell.waveguide import compute_modes
from femwell.maxwell.waveguide import calculate_sfwm_Aeff
from femwell.mesh import mesh_from_OrderedDict
from scipy.constants import c, epsilon_0 ,pi

def n_X(wavelength):
    x = wavelength
    return (1 
            + 2.19244563 / (1 - (0.20746607 / x) ** 2)
            + 0.13435116 / (1 - (0.3985835 / x) ** 2)
            + 2.20997784 / (1 - (0.20747044 / x) ** 2)
    ) ** 0.5

# Box
def n_silicon_dioxide(wavelength):
    x = wavelength
    return (
        1
        + 0.6961663 / (1 - (0.0684043 / x) ** 2)
        + 0.4079426 / (1 - (0.1162414 / x) ** 2)
        + 0.8974794 / (1 - (9.896161 / x) ** 2)
    ) ** 0.5

Clad=1


core = box(0, 0, 0.5, 0.39)
polygons = OrderedDict(
    core=core,
    box=clip_by_rect(core.buffer(1.5, resolution=4), -np.inf, -np.inf, np.inf, 0),
    clad=clip_by_rect(core.buffer(1.5, resolution=4), -np.inf, 0, np.inf, np.inf),
)

resolutions = {"core": {"resolution": 0.025, "distance": 2.}}

mesh = from_meshio(mesh_from_OrderedDict(polygons, resolutions, default_resolution_max=0.6))

num_modes = 2

lambda_p0 = 1.4
lambda_s0 = 1.097
lambda_i0 = 1.686

basis0 = Basis(mesh, ElementTriP0())

epsilon_p = basis0.zeros(dtype=complex)
epsilon_s = basis0.zeros(dtype=complex)
epsilon_i = basis0.zeros(dtype=complex)


for wavelength, epsilon in zip([lambda_p0, lambda_s0, lambda_i0], [epsilon_p, epsilon_s, epsilon_i]):
    for subdomain, n_func in {
        "core": n_X,
        "box": n_silicon_dioxide,
        "clad": lambda _: Clad,
    }.items():
        n = n_func(wavelength)
        epsilon[basis0.get_dofs(elements=subdomain)] = n**2


modes_p = compute_modes(basis0, epsilon_p, wavelength=lambda_p0, num_modes=num_modes, order=1)
modes_s = compute_modes(basis0, epsilon_s, wavelength=lambda_s0, num_modes=num_modes, order=1)
modes_i = compute_modes(basis0, epsilon_i, wavelength=lambda_i0, num_modes=num_modes, order=1)


mode_p = max(modes_p, key=lambda mode: mode.te_fraction)
mode_s = max(modes_s, key=lambda mode: mode.te_fraction)
mode_i = max(modes_i, key=lambda mode: mode.te_fraction)


A_eff = calculate_sfwm_Aeff(basis0, mode_p, mode_s, mode_i)
print(A_eff)

chi_3 = 5e-21  # m^2/V^2  #7e-20?
lambda_p0_m = lambda_p0 * 1e-6  #to m
n_p0 = np.real(mode_p.n_eff)
A_eff_m2 = A_eff * 1e-12  #to m^2

omega_p0 = 2 * pi * c / lambda_p0_m

gamma = (3 * chi_3 * omega_p0) / (4 * epsilon_0 * c**2 * n_p0**2 * A_eff_m2)

print("gamma:",gamma)

After confirming that the electric field distribution was normalized, I obtained a complex number with a negative real part for

$${\iint \mathrm{d} x \mathrm{~d} y u_p(x, y) u_p(x, y) u_s^*(x, y) u_i^*(x, y)}$$

?According to the comparison with Lumerical's results, the Aeff should be on the order of $0.469 \ \text{um}^2$.

The text was updated successfully, but these errors were encountered:

HelgeGehring · 2024-02-23T21:35:16Z

Looks like a great start @Kunyuan-LI !

Three quick things on the first look:

I'd not unfreeze the mode (it's normalized to power=1), but include the factors for the normalization directly in the calculation, i.e. just divide the calculated number by the three normalization factors. There's so many ways to normalize a mode, it might be hard to store all of them 🤔
If we see in the end that we need them more often, we could add it as a cached_property to the mode object 🤔
what do you think?
About the integrals:

@Functional
def E2(w):
   return (w["E"][0] * np.conj(w["E"][0]))

I'm actually a bit confused that this doesn't fail 😅 E[0] should be a 2d vector field. I'd guess what you want to do here is

@Functional
def E2(w):
   return dot(w["E"][0],np.conj(w["E"][0]))

to get the scalar product of the E.

But this also confuses me a bit - the integral you posted in the end doesn't seem to have any vectors inside but uses scalars.
Do we just need there to use the dominant component? (i.e. ['E'][0][0 or 1])
Do you know how that integral is derived?

As long as you use complex epsilons the E-fields' phases will also be complex with a random global phase. While this is nothing to worry about and it can just be ignored, there's no reason for this example to go to complex epsilons.
Just use

epsilon_p = basis0.zeros()
epsilon_s = basis0.zeros()
epsilon_i = basis0.zeros()

and you'll get real fields. While this can still give a random sign to the integral (as the sign of the eigenmode is not defined), we'll get something real :)

Could you make a PR out of this like @elizaleung830 ? That's easiest to discuss/test/improve and in the end we would also get a nice extra example :)

Kunyuan-LI · 2024-02-23T22:37:39Z

Thank you for your answer!

Indeed, the mistake about the normalization of E is quite obvious😅. However, maybe getting it right could be helpful for the subsequent calculations?😂

In the numerical integration of finite elements, for normalization, what we actually consider is the sum of the

$$ E(x,y) \cdot E^*(x,y) $$

calculated separately on each element of the basis. Considering $u_v(x,y)$ where $v=p,s,i$ represents our normalized electric field distribution on the xy-plane, I believe the method of calculation for

$${\iint u_p(x, y) u_p(x, y) u_s^*(x, y) u_i^*(x, y)\, dx\,dy}$$

needs to be modified as well because $u_v(x,y)$ is just a normalized $E(x,y)$ and the product inside should be a three-time dot, then function.assemble?😂

And for the complex epsilons, yes I think you are right. As we just concern about the effective refractive index $n_{eff}$, which is the real part given by femwell for TE and TM modes, so no need to use complex epsilons.

For the PR sure I'll make it ASAP.😄

Kunyuan-LI · 2024-02-24T16:03:52Z

Hello,

I've just made a PR, you can now critique my coding.😂
And for the overlap in SFWM, in this case $E(x,y)$ and $u_v(x,y)$ have a shape of (2318, 3).
So do you have a good idea to multiply them then intergrate them in the xy-plane?🤔

HelgeGehring linked a pull request Mar 4, 2024 that will close this issue

SFWM Aeff calculation #131

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Calculation of effective area for Spontaneous Four-wave Mixing (SFWM) #130

Calculation of effective area for Spontaneous Four-wave Mixing (SFWM) #130

Kunyuan-LI commented Feb 23, 2024 •

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HelgeGehring commented Feb 23, 2024

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Kunyuan-LI commented Feb 24, 2024

Calculation of effective area for Spontaneous Four-wave Mixing (SFWM) #130

Calculation of effective area for Spontaneous Four-wave Mixing (SFWM) #130

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HelgeGehring commented Feb 23, 2024

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