Engineering & Science
Kretschmann Surface Plasmon Resonance (SPR) Biosensor Angle Formula
Calculate the prism-coupled Surface Plasmon Resonance dip angle θ_SPR = DEGREES(ASIN((1 / n_p) * SQRT((|ε_m| * n_d^2) / (|ε_m| - n_d^2)))) for label-free antibody-antigen optical biosensors.
Why free-space lasers CANNOT excite surface plasmons without a Kretschmann prism!
At a gold-water interface (gold real permittivity ε_m = -11.8 at λ = 633 nm, magnitude |ε_m| = 11.8; aqueous buffer refractive index n_d = 1.3330), surface plasmon polaritons carry momentum k_SPP = k_0 * SQRT((|ε_m| * n_d^2) / (|ε_m| - n_d^2)) = 1.4463 * k_0—strictly LARGER than any photon propagating in air or water! Erwin Kretschmann and Heinz Raether (1968) matched that extra wavevector by coupling p-polarized laser light through a high-index glass prism (n_p > k_SPP / k_0) beyond total internal reflection: sin(θ_SPR) = (1 / n_p) * SQRT((|ε_m| * n_d^2) / (|ε_m| - n_d^2))!
Look at the exquisite label-free sensitivity used in Biacore pharmaceutical affinity instruments (BK7 prism n_p = 1.5151, gold |ε_m| = 11.8): pure aqueous buffer (n_d = 1.3330) produces a sharp reflectance dip at θ_SPR = 72.672°; when antibody-antigen binding raises surface refractive index by just +0.0050 RIU to n_d = 1.3380, θ_SPR shifts +0.831° to 73.503° (166°/RIU angular sensitivity)!
| Analyte Channel State (np = 1.5151, Gold |εm| = 11.8) | Effective SPP Index k_SPP/k0 | Resonance Angle θ_SPR (deg) | Biosensor Shift Δθ |
|---|---|---|---|
| Baseline Running Buffer (nd = 1.3330) | 1.4463 (< np) | 72.672° | 0.000° (Reference Dip) |
| Bound Protein Monolayer (nd = 1.3380) | 1.4527 (< np) | 73.503° | +0.831° (+0.005 RIU) |
| Under-Indexed Prism (np <= k_SPP/k0) | sin(θ) >= 1 | "No SPR Coupling (Increase Prism np)" | Requires Sapphire / SF10 Prism |
Kretschmann Prism-Coupled SPR Resonance Angle θ_SPR (degrees)
Verifies plasmon existence (|ε_m| > n_d^2) and prism momentum matching (k_SPP / k_0 < n_p) before evaluating DEGREES(ASIN(...)).
=IF(OR(B2 <= 1, B3 <= 0, B4 <= 0, B3 <= B4 ^ 2), "Invalid Optical Constants", IF(SQRT((B3 * (B4 ^ 2)) / (B3 - (B4 ^ 2))) >= B2, "No SPR Coupling (Increase Prism np)", ROUND(DEGREES(ASIN(SQRT((B3 * (B4 ^ 2)) / (B3 - (B4 ^ 2))) / B2)), 3)))Surface Plasmon Effective Propagation Index n_eff,SPP = k_SPP / k_0
Computes SQRT((|ε_m| * n_d^2) / (|ε_m| - n_d^2)), specifying the minimum prism index n_p required for coupling.
=IF(OR(B3 <= 0, B4 <= 0, B3 <= B4 ^ 2), "Invalid Optical Constants", ROUND(SQRT((B3 * (B4 ^ 2)) / (B3 - (B4 ^ 2))), 4))Try the example
Use the original values or change them to test your understanding.