Transmission Coefficient Calculator

Calculate the transmission coefficient T from amplitude |t| using T = |t|², or find amplitude |t| from a known coefficient input value.

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Transmission Coefficient Formula

The transmission coefficient measures how much of an incident wave successfully passes through a boundary, interface, or potential region. In the most general form, it is defined as transmitted flux divided by incident flux. In many introductory problems, the transmitted and incident waves use the same normalization, so the relation simplifies to the square of the transmission amplitude magnitude.

T = \frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}}

T = \frac{v_{\mathrm{trans}}}{v_{\mathrm{inc}}}\lvert t \rvert^2

\text{If } v_{\mathrm{trans}} = v_{\mathrm{inc}}, \text{ then } T = \lvert t \rvert^2

This calculator uses the matched-normalization case, so entering either the transmission amplitude magnitude |t| or the transmission coefficient T lets you calculate the other directly.

How to Use the Calculator

Enter either the transmission amplitude magnitude |t| or the transmission coefficient T.
Leave the other field blank.
Click Calculate.
Read the computed result.

Reverse Formula

If you already know the transmission coefficient and need the amplitude magnitude in the matched case, take the square root:

\lvert t \rvert = \sqrt{T}

Variable Definitions

T = transmission coefficient, a dimensionless ratio of transmitted flux to incident flux
t = transmission amplitude, generally complex
|t| = magnitude of the transmission amplitude
j_trans = transmitted flux
j_inc = incident flux
v_trans, v_inc = transmitted and incident wave velocities or equivalent normalization factors

What the Result Means

T = 0: no transmission
0 < T < 1: partial transmission
T = 1: complete transmission

In passive, properly normalized systems, T is typically between 0 and 1. A larger value means a larger fraction of the incoming wave is transmitted.

Examples

Example 1: Find transmission coefficient from amplitude magnitude

If the transmission amplitude magnitude is 0.8:

T = \lvert t \rvert^2 = 0.8^2 = 0.64

This means 64% of the normalized incident flux is transmitted.

Example 2: Find amplitude magnitude from transmission coefficient

If the transmission coefficient is 0.25:

\lvert t \rvert = \sqrt{0.25} = 0.5

The corresponding transmission amplitude magnitude is 0.5.

When the Simplified Formula Is Valid

The shortcut T = |t|^2 is valid when the incident and transmitted waves are normalized the same way. That often happens in simplified one-dimensional scattering setups where the wave speeds or wave-number factors on both sides are equal. If they are not equal, the flux ratio must include the extra velocity or normalization term.

T \neq \lvert t \rvert^2 \quad \text{unless the normalization factors match}

Transmission vs. Reflection

Transmission is commonly analyzed together with reflection. For a lossless two-channel system, the transmitted and reflected fractions sum to 1:

R + T = 1

Here, R is the reflection coefficient. If transmission increases, reflection usually decreases, assuming no absorption or gain.

Common Applications

Quantum mechanics and tunneling problems
Wave propagation across boundaries
Optics and layered media
Acoustics and impedance transitions
Electromagnetic wave transmission
General scattering and interface analysis

Common Mistakes

Using T = |t|^2 when the incident and transmitted waves are not equally normalized
Confusing the amplitude t with the coefficient T
Forgetting that t may be complex while the calculator uses only |t|
Assuming all systems must satisfy R + T = 1 even when absorption or gain is present
Entering percentages instead of decimal form, such as using 64 instead of 0.64

Quick Reference

Known Value	Use This Relation	Result
Amplitude magnitude `\|t\|`	$T = \lvert t \rvert^2$	Transmission coefficient
Transmission coefficient `T`	$\lvert t \rvert = \sqrt{T}$	Amplitude magnitude
Unequal normalization	$T = \frac{v_{\mathrm{trans}}}{v_{\mathrm{inc}}}\lvert t \rvert^2$	Flux-corrected transmission

Frequently Asked Questions

Is the transmission coefficient the same as the transmission amplitude?
No. The amplitude is usually a complex quantity, while the transmission coefficient is a real, dimensionless flux ratio. In the matched case, the coefficient equals the square of the amplitude magnitude.

Can the transmission coefficient be greater than 1?
In standard passive normalized systems, it is usually between 0 and 1. Values outside that range typically indicate a different normalization, gain, or a modeling issue.

Why does the calculator ask for amplitude magnitude instead of amplitude?
Because the matched-case relation depends on |t|, not the phase of the complex amplitude.

What if I know transmitted and incident flux directly?
Then compute the coefficient from the flux ratio:

T = \frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}}

Summary of the Calculation

For the matched-normalization case used here, the calculator converts between amplitude magnitude and transmission coefficient using a simple square and square-root relationship. That makes it useful for quick checks in wave transmission, scattering, tunneling, and interface problems where the simplified normalization assumption applies.