SPR Biosensor — Comparative Transfer-Matrix Study

Section IAbstract

A multilayer transfer-matrix engine, cross-validated against the Byrnes tmm reference, evaluates three Kretschmann SPR configurations under identical optical constants and reports their resonance angle, minimum reflectance, angular shift, full-width at half-maximum, and bulk refractive-index sensitivity.

This repository reproduces the comparative analysis of Habib, Roy, Islam, Hassan, Islam & Hossain, Study of Graphene–MoS₂ Based SPR Biosensor with Graphene Based SPR Biosensor: Comparative Approach, International Journal of Natural Sciences Research 7(1), 1–9 (2019), doi:10.18488/journal.63.2019.71.1.9. The implementation is written from first principles using the Fresnel coefficients for p-polarized light, complex Snell propagation, and the Byrnes interface–propagation matrix product. A test suite of sixty-nine cases verifies physical bounds, qualitative trends, and pointwise agreement with an external reference.

Configurations compared

3 layer stacks

Conventional, graphene-enhanced, MoS₂–graphene

Test suite

69 tests

Fresnel, TMM, materials, sensor, metrics, regression

Reference agreement

10⁻¹⁵ abs.

Pointwise vs. Byrnes tmm, machine precision

Headline finding

Using the paper’s stated material constants — SF11 prism n = 1.7786, silver ε = −18.295 + 0.481j, PBS buffer n = 1.34 — the SPR resonance occurs near 52–54°, not the 74–77° reported in the paper. The discrepancy of roughly twenty-two degrees is fundamental and cannot be eliminated by varying film thickness or sensing-medium index. Qualitative trends, namely the angular shift introduced by graphene and the further shift introduced by MoS₂, are reproduced.

Section IIThe Three Configurations

Each stack uses an identical incident prism, an identical silver film, and an identical buffer; the only variable is the bio-recognition overlayer.

Configuration A — Conventional

A bare silver film deposited on an SF11 prism, in contact with phosphate-buffered saline. This is the reference dip used for all subsequent comparisons.

Configuration B — Graphene-enhanced

A single monolayer of graphene is added between the silver film and the buffer. The graphene increases the field overlap with the analyte half-space and shifts the resonance to a slightly higher angle.

Configuration C — MoS₂–Graphene

A monolayer of MoS₂ is inserted between silver and graphene. The stronger optical contrast of MoS₂ in the visible range produces a deeper, broader dip and a larger angular shift, at the cost of resonance width.

Section IIIComputed Results

Resonance metrics from the present implementation, alongside the values originally reported in the paper.

Table 1 · Computed by this repository, paper material constants, PBS sensing medium
Configuration	θ_SPR (°)	R_min	Δθ (°)	FWHM (°)	S (°/RIU)
Conventional (Ag only)	52.67	0.315	0.00	1.11	62.0
Ag + Graphene (1L)	52.80	0.164	0.13	1.26	62.5
Ag + MoS₂ + Graphene	53.49	0.026	0.82	1.87	65.0

Table 2 · Reported in Habib et al. (2019), Table 3
Configuration	θ_SPR (°)	R_min	Δθ (°)
Conventional	74.60	0.3484	0.00
Graphene	74.95	0.1883	0.35
MoS₂–Graphene	76.70	0.0293	2.10

Reading the comparison

The two tables agree on the direction of every effect. Adding graphene reduces R_min and shifts the dip outward; adding MoS₂ deepens the dip further and widens the resonance. The disagreement is in absolute angle, not in trend. The plausible explanations are a different choice of silver optical constants or a different transfer-matrix sign convention in the original work; neither film thickness nor sensing-medium index can move the dip by more than a few degrees in this regime.

Section IVFigures

Reflectance curves and DNA-sensing response, generated directly from the simulation.

Reflectance versus angle for the three configurations at 633 nm. Each curve shows a pronounced minimum near 52 to 54 degrees. — Figure 2Reflectance against incidence angle for the three Kretschmann configurations. The MoS₂–graphene curve shows the deepest minimum and the largest shift relative to the bare silver reference.

DNA sensing response for the graphene configuration at increasing analyte concentration. — Figure 3aGraphene configuration. Resonance shift as a function of DNA concentration, reproduction calibration mode.

DNA sensing response for the MoS₂–graphene configuration at increasing analyte concentration. — Figure 3bMoS₂–graphene configuration. The same protocol applied to the dichalcogenide-augmented stack.

Sensitivity in degrees per refractive index unit for the three configurations. — Figure 4Bulk refractive-index sensitivity, expressed in degrees per refractive index unit, computed from the angular shift between two sensing media bracketing PBS.

Section VParameter Sweeps

Each sweep varies a single design variable while holding all others fixed, isolating its contribution to the resonance.

Resonance angle as a function of silver thickness. — Sweep 1aResonance angle as a function of silver film thickness. The dip is weakly dependent on thickness in the optical-skin-depth regime.

Minimum reflectance as a function of silver thickness, with a clear optimal value. — Sweep 1bMinimum reflectance against silver thickness. The classic critical-coupling minimum appears in the expected window.

Effect of graphene monolayer count on resonance. — Sweep 2Effect of graphene monolayer count. Each additional layer shifts and broadens the dip monotonically.

Effect of MoS₂ monolayer count on resonance. — Sweep 3Effect of MoS₂ monolayer count. Stronger optical contrast yields a larger shift per added layer than graphene alone.

Resonance angle versus sensing medium refractive index, showing a near-linear relationship. — Sweep 4Resonance angle as a function of sensing-medium refractive index. The slope of this line is the bulk sensitivity reported in Table 1.

Section VIMethods & Materials

All computations are performed at a single wavelength of 633 nm with p-polarized illumination. The Fresnel and propagation conventions follow Byrnes, arXiv:1603.02720.

Transfer matrix

The angle-resolved reflectance of an N-layer stack follows from a product of two-by-two interface and propagation matrices applied to the incident field amplitudes.

cos θ_k = √(1 − (n₀/n_k)² sin² θ₀) Snell, complex

r_p = (n_f cos θ_i − n_i cos θ_f) / (n_f cos θ_i + n_i cos θ_f) Fresnel, p

δ_k = 2π n_k d_k cos θ_k / λ Phase

r = M₂₁ / M₁₁, R = |r|² Reflectance

Resonance metrics

The resonance angle is located by parabolic refinement around the discrete minimum. The full-width at half-maximum is taken between the two angles where reflectance equals (1+R_min)/2. Sensitivity is the slope of θ_SPR against sensing-medium refractive index, evaluated by finite difference around n = 1.34.

DNA sensing model

Two operating modes are exposed. The reproduction mode applies an empirical scale factor to recover the angular shifts reported in the paper; this factor is documented and has no independent physical derivation. The physics mode maps molar concentration to a mass concentration and to a refractive-index increment using the bulk dn/dc of DNA, producing values much smaller than the paper implies, consistent with the known limitation that bulk dn/dc does not capture surface accumulation.

Material constants at 633 nm

SF11 prism: n = 1.7786 · Schott catalog
Silver: ε = −18.295 + 0.481j · Johnson & Christy 1972
MoS₂: n = 5.9 + 0.8j · Mak et al., PRL 105, 136805
Graphene: n = 3.0 + 1.1487j · Bruna & Borini, APL 94, 031901
PBS buffer: n = 1.34

Test coverage

Fresnel: Snell, Brewster, total internal reflection, complex media
TMM: Bounds, dip existence, smoothness, layer effects
Cross-validation: Pointwise match against Byrnes tmm
Materials: All constants match paper, all sources cited
Sensor: Stack construction, dimensional analysis
Metrics: Resonance refinement, FWHM, sensitivity, FOM
Regression: Qualitative trends, monotonicity, invariants

Reproducing the figures

git clone https://github.com/ruddro-roy/SPR-Biosensor-Comparative-Approach
cd SPR-Biosensor-Comparative-Approach
pip install -e ".[dev,validation]"

pytest tests/ -v                       # 69 tests
python scripts/reproduce_paper.py      # tables + figures in results/
python scripts/extended_analysis.py    # parameter sweeps

Known limitations

Absolute SPR angles do not match the paper. The qualitative comparative analysis is preserved.
The DNA reproduction mode uses an empirical calibration; the physics mode is consistent with bulk dn/dc values.
A single wavelength of 633 nm is supported, with no spectral dispersion model.
The simulation is fully coherent and assumes flat interfaces; roughness, beam divergence, and incoherent effects are not modeled.

Citation

Habib, M. M., Roy, R., Islam, M. M., Hassan, M., Islam, M. M., & Hossain, M. B. (2019). Study of Graphene–MoS₂ Based SPR Biosensor with Graphene Based SPR Biosensor: Comparative Approach. International Journal of Natural Sciences Research, 7(1), 1–9.

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