Optics 097 · Waveguides, structured light, and modern optics

Surface Plasmon Resonance Lab

How does surface plasmon resonance detect a change in a sample's refractive index? The sample changes the angle at which reflected light becomes darkest. In a prism coupled to a thin metal film, p-polarized light can excite a strong electric field at the metal–sample surface and increase metal absorption. Changing the sample index shifts this resonance. Film thickness and loss also affect the dip, so quantitative sensing needs calibration.

Tune a Kretschmann prism coupler and a finite metal film. Compare p and s polarization, follow the reflection dip as the analyte index changes, inspect the complex electric field and integrate the same field’s Joule loss.

Reflected normal power

R

$0.50$

Power absorbed in metal

A

$50\%$

Analyte-side electric enhancement

F_E

$0.00\pi$

Model regime $\text{valid model regime}$

Exact coherent three-layer plane-wave model at 633 nm, with assumed gold-like permittivity and lossless prism and analyte. Material parameters are illustrative, not fitted to a measured film. Source, collector and curved coupler are schematic and ideally index matched. The field section and film thickness share a scale; electric vectors are enlarged and slowed. No finite beam, grating, molecular binding, roughness, dispersion scan, noise or experimental calibration.

Start with the essentials

Focus question: How does surface plasmon resonance detect a change in a sample's refractive index?
One-sentence intuition: At every angle, $R+T+A=1$ . Above the sample light line there is no normal transmitted power. Near a p-polarized resonance the reflected beam dims and Joule heating in the metal increases; a stronger surface electric field is not an extra source of energy.

Core mathematical model

Finite-film reflection

\begin{gathered}q_j=\sqrt{\varepsilon_j-n_p^2\sin^2\theta}\\r=\frac{r_{01}+r_{12}e^{2ik_0q_1d}}{1+r_{01}r_{12}e^{2ik_0q_1d}}\\r_{ij}=\frac{g_i-g_j}{g_i+g_j},\quad g_j=\begin{cases}q_j&s\\q_j/\varepsilon_j&p\end{cases}\end{gathered}

The scalar amplitude is transverse electric field for s polarization and impedance-normalized transverse magnetic field for p polarization. The normal square root decays into passive outgoing media. Multiple reflections inside the finite film determine the dip; no prescribed resonance line is added. Angles are internal and measured from the interface normal.

Absorption from the electric field

\begin{gathered}F_E=\frac{|\mathbf E(d^+)|^2}{|E_{\rm inc}|^2}\\a(z)=\frac{k_0\operatorname{Im}\varepsilon_m\,|\mathbf E(z)|^2}{n_p\cos\theta\,|E_{\rm inc}|^2}\\A=\int_0^d a(z)\,\mathrm dz\end{gathered}

The enhancement is evaluated on the sample side of the film. Local Joule loss integrates to the same absorption obtained by subtracting reflected and transmitted normal Poynting power. A discontinuous normal electric field at a material boundary does not imply discontinuous normal displacement.

An isolated-interface reference

\begin{gathered}\beta_{\rm sp}=k_0\sqrt{\frac{\varepsilon_m\varepsilon_a}{\varepsilon_m+\varepsilon_a}}\\\ell_{I,z}=\frac1{2\operatorname{Im}k_{za}}\end{gathered}

The isolated-interface mode explains why changing sample index shifts momentum matching. It does not locate the exact finite-film reflection minimum. The displayed dip is found in the computed angular scan. The normal decay length belongs to intensity; field amplitude has twice this decay length. It is undefined as a decay length when the outgoing field propagates.

Common difficulties

Does a dark reflection prove molecular binding?

Typical misconceptionAny reflection dip uniquely identifies molecules on the metal.

Better mental modelA dip depends on angle, film properties, polarization and the surrounding dielectric. Molecular sensing requires a surface-binding model and calibration. Here only a uniform sample refractive index changes; no molecule count or chemical specificity is inferred.

Run the experiment

01
Scene 1: Dark reflection, strong surface field
Drag the cyan source through the dark reflection angle, then compare with the off-resonance preset.
What to observe: Near resonance the reflected collector dims and the sample-side electric vectors grow.
02
Scene 2: Away from resonance
Increase the analyte index and click the shifted gold dip in the scan to select its new angle.
What to observe: A higher sample index shifts the accessible p-polarized dip toward larger internal angles. A dip outside the scan is not reported.
03
Scene 3: Change to s polarization
Choose s polarization, change the actual film thickness, then inspect depth profiles and download the complex field.
What to observe: With s polarization the normal electric component vanishes. Zero loss removes absorption, and zero thickness removes the film entirely.