Optical Concepts

This chapter defines the optical quantities that appear most often in this app and maps them to the relevant fields and result pages.

Scope of this chapter

Complex refractive index and input models

The most basic material quantity is the complex refractive index:

N = n + i k

where:

• n controls phase velocity and refraction;
• k controls absorption and is typically required to satisfy k >= 0.

The current app supports three input models.

For engineering interpretation, prioritize these three rules:

1. Increasing n changes both optical path length and interface contrast, so peak position and peak height may both move.
2. Increasing k increases absorption, which usually raises A and lowers transmission.
3. If dispersion is important, prefer File; otherwise fringe locations and color results can drift from the physical material.

Polarization, pRatio, and incident angle

At oblique incidence, the electric field must be defined relative to the plane of incidence.

Incident Angle is the external incidence angle. In the solver, it changes all of the following:

1. the propagation angle inside each layer;
2. the Fresnel coefficients at each interface;
3. the effective optical path length through each layer.

As a result, changing the angle usually affects both spectral position and contrast, and it strongly affects Psi / Delta. For most introductory cases, start from 0° or a moderate oblique angle such as 60° ~ 70°.

Using Snell's law to interpret refraction through the stack

Use the following relation to interpret refraction through the stack:

n_i sin(theta_i) = n_j sin(theta_j)

In the context of this app, that means:

The app does not ask you to enter internal layer angles manually. The important point is that changing Incident Angle changes the propagation condition of the full stack, not just the first interface.

Cone angle averaging

Standard TMM assumes ideal plane-wave incidence, but real optical systems use convergent or divergent beams with a finite angular spread. Cone angle averaging models this by computing results for multiple rays distributed over a cone centered on the nominal incident angle and returning the weighted average.

Input modes and conversion formulas

The software accepts three ways to specify the cone size. All are converted internally to the half-cone angle θ_cone.

Here n_medium is the refractive index of the incidence medium. When the incidence medium is air (n ≈ 1), NA is equivalent to sin(θ_cone).

Ring sampling

Discrete sampling within the cone uses a concentric-ring layout:

• 1 central ray at the nominal incident angle
• 12 equally spaced rays per ring

Effective ray count = 1 + 12 × ringCount

ringCount ranges from 2 to 20, giving 25 to 241 effective rays. Increasing ringCount improves angular integration accuracy at linearly increasing computational cost.

Angular distribution weights

Scope of applicability

Cone angle averaging applies only to certain result types:

Unsupported result types are still computed at the single nominal incident angle.

Validity constraint

The cone must not extend to grazing incidence:

incidentAngle + θ_cone < 89.9°

If this condition is violated, the software reports a validation error and blocks computation.

Birefringence and extended parameters

Birefringent materials have different complex refractive indices along different principal directions.

In the current app, birefringence can be represented in two ways:

1. Constant Birefringence: enter one set of ordinary and extraordinary constants directly;
2. File: import wavelength-dependent birefringent data.

The implementation limits are explicit:

• surrounding media do not expose birefringent parameters;
• incoherent layers cannot use the constant birefringent model;
• incoherent layers also cannot use birefringent refractive-index files.

So in practice, birefringence is intended for coherent thin-film modeling, not for thick-substrate or incoherent approximations.

Physical definition of Psi and Delta

Ellipsometric outputs are not independent quantities. They are derived from the ratio of reflection coefficients:

• tan(Psi) = |r_p / r_s|
• Delta = arg(r_p) - arg(r_s)

This gives the two curves distinct roles:

In engineering work, Psi / Delta are commonly used for thickness and refractive-index inversion. The app follows the same physical requirement: the stack must remain coherent. If any enabled layer is incoherent, Psi / Delta are blocked.

Mapping the concepts to UI fields