Calculator Inputs
Example Data Table
These are typical inputs and expected trends. Your results depend on the chosen model.
| Scenario | Inputs | Typical Output |
|---|---|---|
| Fresnel | n1=1.0, n2=1.5, θi=30°, TE | T ≈ 0.942 (power) |
| Impedance | Z1=415, Z2=1500 | T ≈ 0.702 (power) |
| Quantum | E=0.20 eV, V0=0.35 eV, a=2 nm, m*=1 | T typically < 1, often small |
Formulas Used
1) Interface impedances (normal incidence)
For two lossless media with real impedances Z1 and Z2:
- r = (Z2 − Z1) / (Z2 + Z1)
- R = r²
- T = 4 Z1 Z2 / (Z1 + Z2)²
- t = 2 Z2 / (Z1 + Z2)
2) Fresnel transmission (angle + polarization)
Uses Snell’s law n1 sinθi = n2 sinθt and amplitude coefficients.
- TE: t = 2 n1 cosθi / (n1 cosθi + n2 cosθt)
- TM: t = 2 n1 cosθi / (n2 cosθi + n1 cosθt)
- T = (n2 cosθt)/(n1 cosθi) · t²
3) Quantum rectangular barrier (1D)
Transmission probability through a barrier of width a and height V0:
- E < V0: T = 1 / (1 + (V0² sinh²(κa)) / (4E(V0−E)))
- E > V0: T = 1 / (1 + (V0² sin²(qa)) / (4E(E−V0)))
How to Use This Calculator
- Select the scenario that matches your physical setup.
- Enter parameters using consistent units shown in each field.
- Press Compute to display results above the form.
- Check T for transmitted power or probability, and R for reflection.
- Use the export buttons to save CSV or PDF for your notes.
Professional Notes and Interpretation
1) Meaning of T in this calculator
The transmission coefficient T reports how much incident flux crosses a boundary. In the Fresnel and impedance scenarios it is a power (intensity) ratio. In the quantum scenario it is a transmission probability for a 1D rectangular barrier.
2) Fresnel optics: angle and polarization
For light at a dielectric interface, T depends on incidence angle and polarization (TE or TM). For air to glass, typical refractive indices are n1=1.00 and n2=1.50. At 30°, TE power transmission is commonly near 0.94 for a single boundary, while TM can differ noticeably.
3) Critical angle and total internal reflection
If n1 > n2, Snell's law gives a critical angle θc=asin(n2/n1). Beyond θc, the transmitted field is evanescent and the calculator reports total internal reflection. For glass-to-air (1.50 to 1.00), θc is about 41.8°.
4) Impedance interfaces: fast engineering estimate
Many wave problems reduce to normal-incidence impedance matching. The model uses real impedances and gives a power transmission T=4Z1Z2/(Z1+Z2)^2. Typical acoustic impedances are roughly 0.0004 to 0.0005 MRayl for air and 1.4 to 1.6 MRayl for water, explaining strong reflection at the air-water surface.
5) Matching layers and practical design
High transmission requires impedances that are close in value. When a direct match is impossible, an intermediate matching layer can reduce reflections. In optics, thin-film coatings are used to lower reflection at a target wavelength. In ultrasonics, coupling gels and matching layers improve energy transfer from the transducer into tissue.
6) Quantum barrier trends: width and height
For E < V0, transmission decreases rapidly as width increases because the hyperbolic term grows with κa. With the example E=0.20 eV and V0=0.35 eV, increasing a from 2 to 3 nm often reduces T by a large factor, even when all other inputs stay fixed.
7) Effective mass and de Broglie scale
The quantum model uses an effective mass m* through the wave numbers. Many semiconductors have m*/me between about 0.05 and 0.5. Smaller effective mass generally increases the de Broglie wavelength and can increase tunneling for the same barrier geometry.
8) Sanity checks and reporting
For lossless Fresnel and impedance cases, expect 0 ≤ T ≤ 1 and 0 ≤ R ≤ 1. A helpful check is that T+R should be close to 1 for a single, lossless boundary. In reports, record the scenario, all inputs, and exported CSV or PDF results for traceability.
FAQs
1) Which scenario should I choose?
Use Fresnel for light with angle and polarization. Use Impedance for normal-incidence power transfer with real impedances. Use Quantum for 1D transmission through a rectangular potential barrier.
2) What is the difference between amplitude and power transmission?
Amplitude coefficients describe field ratios and can depend on conventions. Power transmission T is the conserved transmitted-to-incident power ratio for lossless cases, and is usually the quantity reported in experiments.
3) Why does the Fresnel model show total internal reflection?
When n1 > n2 and the incident angle exceeds θc, the transmitted wave becomes evanescent. Power transmission across a single boundary approaches zero while reflectance approaches one.
4) Can power transmission be greater than 1?
For lossless single interfaces, power transmission is bounded between 0 and 1. Values outside this range usually indicate invalid inputs, unit mistakes, or that the assumptions (lossless, real parameters) do not apply.
5) What impedance should I use for acoustics?
Use acoustic impedance Z=ρc in Rayl (Pa·s/m). As long as both media use consistent units, the interface formulas apply. Large impedance mismatches imply strong reflection and small transmitted power.
6) How accurate is the quantum barrier result?
It is an ideal 1D rectangular-barrier model. It captures main trends with energy, width, and effective mass, but ignores phonons, multidimensional effects, image forces, and detailed band structure.
7) What should I include in a lab note?
Record the scenario, all parameter values, and the resulting T and R. Note assumptions (lossless media, normal incidence, effective mass choice). Attach the CSV or PDF export for reproducibility.