### Install femwell using pip

Source: https://github.com/helgegehring/femwell/blob/main/docs/install.md

Use this command to install the femwell package via pip. Ensure pip is up-to-date.

```bash
pip install femwell
```

--------------------------------

### Coulomb/Electrostatic Solver Setup

Source: https://context7.com/helgegehring/femwell/llms.txt

Set up geometry for solving electrostatic potential distributions in capacitor and electrode analysis. This snippet focuses on defining the geometry and mesh.

```python
from femwell.coulomb import solve_coulomb
from shapely.geometry import box

# Define parallel plate capacitor geometry
electrode_left = box(-6, 0.5, -3, 2.3)
electrode_right = box(3, 0.5, 6, 2.3)
slab = box(-10, 0, 10, 0.5)
env = slab.buffer(20, resolution=8)

polygons_cap = OrderedDict(
    electrode_left=electrode_left,
    electrode_right=electrode_right,
    slab=slab,
    env=env,
)

resolutions_cap = dict(
    slab={"resolution": 0.1, "distance": 0.1},
)

mesh_cap = from_meshio(
    mesh_from_OrderedDict(polygons_cap, resolutions_cap, default_resolution_max=5)
)
```

--------------------------------

### Install slepc4py for slepc eigensolver

Source: https://github.com/helgegehring/femwell/blob/main/docs/install.md

Install slepc4py with complex support using conda, required for the slepc eigensolver. This command ensures the correct version for complex number support.

```bash
conda install slepc4py=*=complex*
```

--------------------------------

### Static Thermal Simulation

Source: https://context7.com/helgegehring/femwell/llms.txt

Solve steady-state heat distribution with Joule heating sources. Requires defining geometry, mesh, material properties, and current densities.

```python
from femwell.thermal import solve_thermal
from shapely.geometry import LineString

# Define geometry with heater element
polygons_thermal = OrderedDict(
    bottom=LineString([(-8, -1), (8, -1)]),  # Fixed temperature boundary
    core=Polygon([(-0.25, 0), (-0.25, 0.22), (0.25, 0.22), (0.25, 0)]),
    heater=Polygon([(-1, 2), (-1, 2.14), (1, 2.14), (1, 2)]),
    clad=box(-8, 0, 8, 3),
    box=box(-8, -1, 8, 0),
)

resolutions_thermal = dict(
    core={"resolution": 0.04, "distance": 1},
    heater={"resolution": 0.1, "distance": 1},
)

mesh_thermal = from_meshio(
    mesh_from_OrderedDict(polygons_thermal, resolutions_thermal, default_resolution_max=0.6)
)

# Define thermal conductivity (W/m·K, scaled for um units)
basis0_thermal = Basis(mesh_thermal, ElementTriP0(), intorder=4)
thermal_conductivity = basis0_thermal.zeros()
for domain, value in {
    "core": 148,    # Silicon
    "box": 1.38,    # SiO2
    "clad": 1.38,   # SiO2
    "heater": 28,   # TiN
}.items():
    thermal_conductivity[basis0_thermal.get_dofs(elements=domain)] = value
thermal_conductivity *= 1e-12  # Convert to um^-2 units

# Solve thermal distribution with current density
current_density = 0.007 / polygons_thermal["heater"].area  # A/um^2

basis_T, temperature = solve_thermal(
    basis0_thermal,
    thermal_conductivity,
    specific_conductivity={"heater": 2.3e6},  # S/m
    current_densities={"heater": current_density},
    fixed_boundaries={"bottom": 0},  # Reference temperature
)

# Plot temperature distribution
basis_T.plot(temperature, shading="gouraud", colorbar=True)
plt.title("Temperature Distribution (K above ambient)")
plt.show()
```

--------------------------------

### Calculate Mode Properties and Metrics

Source: https://context7.com/helgegehring/femwell/llms.txt

Extract physical properties from computed modes such as confined power, confinement factor, effective area, and propagation loss.

```python
import numpy as np

mode = modes[0]  # First (fundamental) mode

# Calculate power confined in a specific region
power_in_core = mode.calculate_power(elements="core")
print(f"Power in core: {np.real(power_in_core):.4f}")

# Calculate confinement factor
confinement = mode.calculate_confinement_factor(elements="core")
print(f"Confinement factor: {np.real(confinement):.4f}")

# Calculate effective area (for nonlinear optics)
A_eff = mode.calculate_effective_area(field="xy")
print(f"Effective area: {A_eff:.4f} um^2")

# Calculate propagation loss over a distance (for lossy materials)
# Requires complex epsilon with imaginary part
distance = 1000  # micrometers
loss_dB = mode.calculate_propagation_loss(distance)
print(f"Propagation loss over {distance}um: {loss_dB:.2f} dB")

# Access wavelength and frequency
print(f"Wavelength: {mode.wavelength:.4f} um")
print(f"Frequency: {mode.frequency:.4e} Hz")
```

--------------------------------

### Define Permittivity and Solve Coulomb Equation

Source: https://context7.com/helgegehring/femwell/llms.txt

Sets up permittivity for a layered dielectric and solves the Coulomb equation to find the electric potential. Requires mesh and basis definitions.

```python
from skfem import Basis, ElementTriP0
import matplotlib.pyplot as plt

# Define permittivity
basis_cap = Basis(mesh_cap, ElementTriP0())
epsilon = basis_cap.zeros() + 3.9  # Air/oxide
epsilon[basis_cap.get_dofs(elements="slab")] = 28.4  # High-k dielectric

# Solve Coulomb equation with boundary conditions
basis_u, potential = solve_coulomb(
    basis_cap,
    epsilon,
    {
        "electrode_left___slab": 1.0,   # 1V on left electrode
        "electrode_right___slab": 0.0,  # 0V on right electrode
    },
)

# Plot electric potential
basis_u.plot(potential, shading="gouraud", colorbar=True)
plt.title("Electric Potential (V)")
plt.show()
```

--------------------------------

### Transient Thermal Simulation

Source: https://context7.com/helgegehring/femwell/llms.txt

Simulate time-dependent thermal behavior for phase shifter response analysis. Requires defining thermal diffusivity and time-varying current.

```python
from femwell.thermal_transient import solve_thermal_transient

# Define thermal diffusivity (m^2/s, scaled for um units)
thermal_diffusivity = basis0_thermal.zeros()
for domain, value in {
    "heater": 28 / 598 / 5240,     # TiN: k/(c*rho)
    "box": 1.38 / 709 / 2203,      # SiO2
    "clad": 1.38 / 709 / 2203,
    "core": 148 / 711 / 2330,      # Silicon
}.items():
    thermal_diffusivity[basis0_thermal.get_dofs(elements=domain)] = value
thermal_diffusivity *= 1e12  # Convert to um^2 units

# Define time-varying current
dt = 0.1e-5  # Time step in seconds
steps = 100
heater_area = polygons_thermal["heater"].area
current = lambda t: 0.007 / heater_area * ((t < dt * steps / 10) + (t > dt * steps / 2))

# Run transient simulation
basis_T, temperatures = solve_thermal_transient(
    basis0_thermal,
    thermal_conductivity,
    thermal_diffusivity,
    specific_conductivity={"heater": 2.3e6},
    current_densities_0={"heater": current(0)},
    current_densities={"heater": current},
    fixed_boundaries={"bottom": 0},
    dt=dt,
    steps=steps,
)

# Plot temperature evolution
times = np.array([dt * i for i in range(steps + 1)])
avg_temps = [np.mean(T) for T in temperatures]
plt.plot(times * 1e6, avg_temps)
plt.xlabel("Time (us)")
plt.ylabel("Average Temperature (K)")
plt.show()
```

--------------------------------

### Compute Modes for Bent Waveguide Analysis

Source: https://context7.com/helgegehring/femwell/llms.txt

Calculates the modes of a waveguide with a finite bend radius. Requires basis, permittivity, wavelength, and bend radius.

```python
from skfem import Basis, ElementTriP0

# Compute modes with finite bend radius
radius = 10  # Bend radius in um

modes_bent = compute_modes(
    basis0,
    epsilon,
    wavelength=1.55,
    num_modes=1,
    radius=radius  # Enable bent waveguide analysis
)

# Compare effective index with straight waveguide
mode_straight = modes[0]
mode_bent = modes_bent[0]

print(f"Straight waveguide n_eff: {mode_straight.n_eff:.6f}")
print(f"Bent waveguide n_eff (R={radius}um): {mode_bent.n_eff:.6f}")

# Calculate bending loss
loss_per_um = mode_bent.calculate_propagation_loss(1)
circumference = 2 * np.pi * radius
loss_per_turn = loss_per_um * circumference
print(f"Loss per 90-degree bend: {loss_per_turn/4:.4f} dB")
```

--------------------------------

### Calculate Coupling Coefficients

Source: https://context7.com/helgegehring/femwell/llms.txt

Use this to calculate coupling coefficients between waveguide modes for directional coupler design. Requires defining waveguide geometries and calculating modes.

```python
from scipy.constants import epsilon_0, speed_of_light

# Define two waveguides in a coupled system
gap = 0.4
w_core_1, w_core_2 = 0.45, 0.46

polygons_coupled = OrderedDict(
    core_1=Polygon([
        (-w_core_1 - gap/2, 0), (-w_core_1 - gap/2, 0.22),
        (-gap/2, 0.22), (-gap/2, 0),
    ]),
    core_2=Polygon([
        (gap/2, 0), (gap/2, 0.22),
        (w_core_2 + gap/2, 0.22), (w_core_2 + gap/2, 0),
    ]),
    clad=box(-2, -1, 2, 1),
    box=box(-2, -1, 2, 0),
)

# Calculate modes with both waveguides present
epsilon_both = basis0.zeros() + 1.444**2
epsilon_both[basis0.get_dofs(elements=("core_1", "core_2"))] = 3.4777**2
modes_coupled = compute_modes(basis0, epsilon_both, wavelength=1.55, num_modes=2)

# Calculate coupling length for power transfer
n_eff_1, n_eff_2 = modes_coupled[0].n_eff, modes_coupled[1].n_eff
coupling_length = wavelength / (2 * np.abs(n_eff_1 - n_eff_2))
print(f"Coupling length for full power transfer: {coupling_length:.2f} um")
```

--------------------------------

### Visualize Schrödinger Equation Bound States

Source: https://github.com/helgegehring/femwell/blob/main/docs/math/schroedinger.md

Plots the conditions for symmetric and asymmetric bound states of the Schrödinger equation in a potential well. Requires numpy and matplotlib. The plot visualizes the relationship between energy levels and well parameters.

```python
import numpy as np
import matplotlib.pyplot as plt

u0 = np.sqrt(20)
v = np.linspace(0,5,10000)
yc = np.sqrt(u0**2-v**2)
plt.ylim(0,u0+1)
plt.plot(v, yc)

y = v*np.tan(v)
y = np.where(y>-10,y,np.nan)
plt.plot(v, y, label='symmetric')
idx_s = np.argwhere(np.nan_to_num(np.diff(np.sign(y - yc)),-1))[:,0]
plt.plot(v[idx_s], y[idx_s], 'ro')
print(v[idx_s])

y = -v*1/np.tan(v)
plt.plot(v, np.where(y>-10,y,np.nan), label='asymmetric')
idx_a = np.argwhere(np.nan_to_num(np.diff(np.sign(y - yc)),-1))[:,0]
plt.plot(v[idx_a], y[idx_a], 'ro')
print(v[idx_a])

plt.legend()
plt.show()
```

--------------------------------

### Compute Optical Waveguide Eigenmodes

Source: https://context7.com/helgegehring/femwell/llms.txt

Calculate effective refractive index, TE/TM fractions, and field distributions for waveguide modes. Requires a defined permittivity distribution based on the mesh subdomains.

```python
from skfem import Basis, ElementTriP0
from femwell.maxwell.waveguide import compute_modes

# Create basis for piecewise constant permittivity
basis0 = Basis(mesh, ElementTriP0())

# Define permittivity distribution (epsilon = n^2)
epsilon = basis0.zeros()
for subdomain, n in {"core": 3.4777, "box": 1.444, "clad": 1.0}.items():
    epsilon[basis0.get_dofs(elements=subdomain)] = n**2

# Compute modes at specified wavelength
wavelength = 1.55  # micrometers
modes = compute_modes(
    basis0,
    epsilon,
    wavelength=wavelength,
    num_modes=3,  # Number of modes to compute
    order=2       # Basis function order (1 or 2)
)

# Access mode properties
for i, mode in enumerate(modes):
    print(f"Mode {i}: n_eff = {mode.n_eff:.6f}")
    print(f"  TE fraction: {mode.te_fraction:.3f}")
    print(f"  TM fraction: {mode.tm_fraction:.3f}")

# Visualize electric field components
modes[0].show("E", part="real", colorbar=True)

# Plot mode intensity
modes[0].show("I", colorbar=True)
```

--------------------------------

### Plot Subdomain Boundaries

Source: https://context7.com/helgegehring/femwell/llms.txt

Highlights only the boundaries of subdomains in a mesh, useful for verifying subdomain definitions without showing the entire mesh.

```python
from femwell.visualization import plot_subdomain_boundaries
import matplotlib.pyplot as plt

# Plot subdomain boundaries only
plot_subdomain_boundaries(mesh)
plt.title("Subdomain Boundaries")
plt.show()
```

--------------------------------

### Generate 2D Meshes from Shapely Polygons

Source: https://context7.com/helgegehring/femwell/llms.txt

Create a 2D mesh from defined Shapely polygons with localized resolution control. Requires an OrderedDict to define geometry priority and a resolution dictionary for mesh refinement.

```python
from collections import OrderedDict
from shapely.geometry import Polygon, box
from shapely.ops import clip_by_rect
import numpy as np
from skfem.io.meshio import from_meshio
from femwell.mesh import mesh_from_OrderedDict

# Define waveguide geometry using Shapely
wg_width = 0.5
wg_thickness = 0.22
core = box(-wg_width / 2, 0, wg_width / 2, wg_thickness)
env = core.buffer(2, resolution=8)

# Create ordered dictionary of polygons (order determines override priority)
polygons = OrderedDict(
    core=core,
    box=clip_by_rect(env, -np.inf, -np.inf, np.inf, 0),
    clad=clip_by_rect(env, -np.inf, 0, np.inf, np.inf),
)

# Define mesh resolution: finer near the core, coarser elsewhere
resolutions = dict(
    core={"resolution": 0.03, "distance": 0.5}  # 30nm resolution, 500nm falloff
)

# Generate mesh and convert to scikit-fem format
mesh = from_meshio(
    mesh_from_OrderedDict(
        polygons,
        resolutions,
        filename="mesh.msh",
        default_resolution_max=0.5
    )
)

# Visualize the mesh
mesh.draw().show()
```

--------------------------------

### Compute Mode Overlap Integrals

Source: https://context7.com/helgegehring/femwell/llms.txt

Calculate overlap integrals between modes to analyze coupling and mode matching. Useful for generating an overlap matrix to verify mode orthonormality.

```python
import numpy as np

# Calculate overlap integral between two modes
overlap = modes[0].calculate_overlap(modes[1])
print(f"Overlap between mode 0 and mode 1: {np.abs(overlap):.6f}")

# Build overlap matrix for all modes
num_modes = len(modes)
overlap_matrix = np.zeros((num_modes, num_modes), dtype=complex)
for i in range(num_modes):
    for j in range(num_modes):
        overlap_matrix[i, j] = modes[i].calculate_overlap(modes[j])

print("Overlap matrix:")
print(np.abs(overlap_matrix))

# Modes should be orthonormal (diagonal ~1, off-diagonal ~0)
```

--------------------------------

### Plot Mesh Domains

Source: https://context7.com/helgegehring/femwell/llms.txt

Visualizes the different domains within a mesh, useful for debugging mesh generation and subdomain definitions.

```python
from femwell.visualization import plot_domains
import matplotlib.pyplot as plt

# Plot mesh with colored domains
plot_domains(mesh)
plt.title("Mesh Domains")
plt.show()
```

--------------------------------

### Calculate Electric Field

Source: https://context7.com/helgegehring/femwell/llms.txt

Calculates the electric field components from the solved electric potential and permittivity. Requires the electric potential and basis.

```python
from skfem import ElementDG

# Calculate electric field
E_x = basis_u.project(-basis_cap.interpolate(epsilon) * basis_u.interpolate(potential).grad[0])
basis_u.plot(E_x, shading="gouraud", colorbar=True)
plt.title("Electric Field E_x")
plt.show()
```

--------------------------------

### Calculate Fiber Overlap Integral

Source: https://context7.com/helgegehring/femwell/llms.txt

Computes the coupling efficiency between a waveguide mode and a Gaussian fiber mode using the overlap integral. Requires mode data and fiber parameters.

```python
from femwell.fiber import e_field_gaussian, overlap
import numpy as np

# Define fiber parameters
mfr = 5.2  # Mode field radius in um
fiber_n = 1.0  # Refractive index
wavelength = 1.55

# Get waveguide mode field
(Ex, Ey), Ez = modes[0].basis.interpolate(modes[0].E)

# Create Gaussian fiber field on same basis
# The fiber field is evaluated at mesh quadrature points
fiber_E = e_field_gaussian(
    r=np.sqrt(modes[0].basis.X[0]**2 + modes[0].basis.X[1]**2),
    z=0,
    mfr=mfr,
    refractive_index=fiber_n,
    wavelength=wavelength
)

# Calculate overlap integral
fiber_coupling = overlap(modes[0].basis, Ex, fiber_E)
print(f"Fiber coupling efficiency: {fiber_coupling**2:.4f}")
```

--------------------------------

### Plot Mode Field Components

Source: https://context7.com/helgegehring/femwell/llms.txt

Visualizes individual components (Ex, Ey, Ez) of a mode's electric field. Requires a computed mode object.

```python
import matplotlib.pyplot as plt

# Plot mode field components individually
mode = modes[0]
fig, axs = plt.subplots(1, 3, figsize=(12, 4))
for ax, comp in zip(axs, ["x", "y", "z"]):
    mode.plot_component("E", comp, part="real", ax=ax, colorbar=True)
plt.tight_layout()
plt.show()
```

--------------------------------

### Calculate Coaxial Cable Impedance

Source: https://context7.com/helgegehring/femwell/llms.txt

Computes the characteristic impedance of a coaxial cable using mode analysis. Requires defining the cable geometry, dielectric properties, and frequency.

```python
import shapely
from skfem import Functional, Basis, ElementTriP0
from skfem.helpers import inner
import scipy.constants
from collections import OrderedDict
from femwell.mesh import from_meshio, mesh_from_OrderedDict

# Define coaxial cable geometry
radius_inner = 0.512e-3  # Inner conductor radius (m)
radius_outer = 2.23e-3   # Outer conductor radius (m)

core = shapely.Point(0, 0).buffer(radius_inner)
isolator = shapely.Point(0, 0).buffer(radius_outer)

polygons_coax = OrderedDict(
    surface=shapely.LineString(isolator.exterior),
    surface_core=shapely.LinearRing(core.exterior),
    isolator=isolator - core,
)

resolutions_coax = dict(isolator={"resolution": 0.05, "distance": 1})

mesh_coax = from_meshio(
    mesh_from_OrderedDict(polygons_coax, resolutions_coax, default_resolution_max=1)
)

# Set up dielectric
basis_coax = Basis(mesh_coax, ElementTriP0(), intorder=4)
epsilon_coax = basis_coax.zeros() + 1.29  # Polyethylene dielectric

# Compute TEM mode with metallic boundaries
frequency = 10e9  # 10 GHz
modes_coax = compute_modes(
    basis_coax,
    epsilon_coax,
    wavelength=scipy.constants.speed_of_light / frequency,
    num_modes=1,
    metallic_boundaries=("surface", "surface_core")
)

print(f"Propagation constant: {1/modes_coax.n_effs[0]:.6f}")

# Calculate current for impedance
@Functional(dtype=np.complex64)
def current_form(w):
    return inner(np.array([w.n[1], -w.n[0]]), w.H)

(ht, ht_basis), (hz, hz_basis) = modes_coax[0].basis.split(modes_coax[0].H)
facet_basis = ht_basis.boundary(facets=mesh_coax.boundaries["isolator___isolator"])
current = abs(current_form.assemble(facet_basis, H=facet_basis.interpolate(ht)))

# Characteristic impedance Z0 = V/I (normalized to 1V)
Z0 = 1 / current
print(f"Characteristic impedance: {Z0:.1f} Ohms")
```

=== COMPLETE CONTENT === This response contains all available snippets from this library. No additional content exists. Do not make further requests.