SPR Biosensor Comparative Approach

Transfer-matrix simulation of Graphene and MoS₂-Graphene SPR biosensors

Companion site: ruddro-roy.github.io/SPR-Biosensor-Comparative-Approach — a publication-style summary of the computed results, figures, parameter sweeps, and methods.

A research-grade computational optics codebase implementing the Kretschmann-configuration Surface Plasmon Resonance (SPR) simulation from:

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. DOI: 10.18488/journal.63.2019.71.1.9

What This Repository Does

• Implements a multilayer transfer-matrix method (TMM) for p-polarized angular interrogation at 633 nm
• Compares three Kretschmann SPR configurations:
  1. Conventional: Prism / Ag / Sensing medium
  2. Graphene-enhanced: Prism / Ag / Graphene / Sensing medium
  3. MoS₂-Graphene: Prism / Ag / MoS₂ / Graphene / Sensing medium
• Generates publication-quality figures and metrics tables
• Provides parameter sweeps over Ag thickness, layer counts, and sensing-medium RI
• Includes a comprehensive test suite validated against the Byrnes tmm package

Key Scientific Finding

The TMM implementation in this repository is validated against the Byrnes tmm package (arXiv:1603.02720) to machine precision. Using the paper's stated material constants (SF11 n=1.7786, Ag ε=−18.295+0.481j at 633 nm, PBS n=1.34), the SPR resonance occurs at approximately 52–54°, not the 74–77° reported in the paper.

This ~22° discrepancy is fundamental and cannot be resolved by adjusting film thickness or sensing-medium RI. It indicates the paper likely used different Ag optical constants than those stated, or a different TMM convention. The qualitative trends (graphene shifts SPR to higher angle, MoS₂+graphene shifts further, sensitivity increases with 2D layers) are correctly reproduced.

Computed Results (this repo)

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

Paper-reported values (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

Physics

Transfer Matrix Method

The simulation computes p-polarized reflectance for an N-layer stack using:

1. Snell's law generalized for complex refractive indices: cos(θ_k) = √(1 − (n₀/n_k)² sin²(θ₀))

2. Fresnel coefficients at each interface: r_p = (n_f cos θ_i − n_i cos θ_f) / (n_f cos θ_i + n_i cos θ_f)

3. Propagation phase through each film: δ_k = 2π n_k d_k cos(θ_k) / λ

4. Transfer matrix product using interface + propagation matrices (Byrnes convention)

5. Reflection coefficient: r = M₂₁/M₁₁, R = |r|²

Material Constants (at 633 nm)

Material	Refractive Index	Source
SF11 prism	1.7786	Schott catalog
Silver (Ag)	√(−18.295 + 0.481j) ≈ 0.056 + 4.278j	Johnson & Christy (1972)
MoS₂	5.9 + 0.8j	Mak et al., PRL 105 (2010)
Graphene	3.0 + 1.1487j	Bruna & Borini, APL 94 (2009)
PBS buffer	1.34	Standard visible-range value

DNA Sensing Model

Two modes are provided:

• Reproduction mode: Uses an inverse-calibrated scale factor to match the paper's reported angular shifts. The formula δn = dn/dc × c_total × 1.87×10⁻⁵ has no independent physical derivation.

• Physics mode: Maps nM concentration → mass concentration → δn using: c_mass = c_nM × MW × 10⁻¹² g/cm³, then δn = (dn/dc) × c_mass. This gives δn ~ 10⁻⁶, much smaller than the paper implies, consistent with the known limitation that bulk dn/dc does not capture surface accumulation effects.

Installation

# Clone and install
git clone <repo-url>
cd SPR-Biosensor-Comparative-Approach
pip install -e ".[dev]"

# Optional: install Byrnes tmm for cross-validation
pip install -e ".[validation]"

Requirements: Python ≥ 3.9, NumPy, SciPy, Matplotlib

Usage

# Run all tests
pytest tests/ -v

# Reproduce paper results and generate figures
python scripts/reproduce_paper.py

# Run extended parameter sweeps
python scripts/extended_analysis.py

All output figures are saved to results/.

Repository Structure

├── src/spr/                  # Main package
│   ├── optics/
│   │   ├── fresnel.py        # Fresnel coefficients and Snell's law
│   │   └── tmm.py            # Transfer Matrix Method engine
│   ├── materials/
│   │   └── database.py       # Optical constants database
│   ├── models/
│   │   └── sensor.py         # Layer stack builder, DNA models
│   ├── analysis/
│   │   └── metrics.py        # Resonance finder, FWHM, sensitivity, FOM
│   └── plotting/
│       └── figures.py        # Publication-quality figure generation
├── tests/                    # 69 tests
│   ├── test_fresnel.py       # Fresnel coefficient tests
│   ├── test_tmm.py           # TMM tests + Byrnes cross-validation
│   ├── test_materials.py     # Material database tests
│   ├── test_sensor.py        # Layer stack + DNA model tests
│   ├── test_metrics.py       # Metrics analysis tests
│   └── test_paper_regression.py  # Physics regression tests
├── scripts/
│   ├── reproduce_paper.py    # Paper reproduction + tables + figures
│   └── extended_analysis.py  # Parameter sweeps
├── results/                  # Generated figures (from actual runs)
├── pyproject.toml            # Package configuration
└── README.md

Tests

The test suite covers:

• Fresnel physics: Snell's law, Brewster angle, total internal reflection, complex media
• TMM correctness: Reflectance bounds, SPR dip existence, smoothness, layer effects
• Cross-validation: Machine-precision match against Byrnes tmm package at multiple angles
• Materials: All constants match paper values, all sources documented
• Sensor models: Layer stack construction, DNA RI models, dimensional analysis
• Metrics: Resonance refinement, FWHM, sensitivity, FOM
• Physics regression: Qualitative trends, physical invariants, monotonicity

Known Limitations

1. Absolute SPR angles do not match the paper — see Key Scientific Finding above. The qualitative comparative analysis is preserved.

2. DNA sensing model is approximate — the reproduction mode uses an empirical calibration factor; the physics mode gives much smaller RI changes than the paper implies, because bulk dn/dc does not capture surface accumulation.

3. Fixed wavelength only — material constants are provided at 633 nm only. Spectral dispersion models are not included due to lack of sourced data.

4. Coherent TMM only — assumes coherent light and perfectly flat interfaces. Roughness, incoherent effects, and beam divergence are not modeled.

5. dn/dc for DNA — the value 0.182 cm³/g is an approximation. Actual dn/dc varies with DNA sequence, ionic strength, and surface coverage conditions.

References

• Habib et al. (2019), Int. J. Natural Sciences Research, 7(1), 1–9
• Byrnes, "Multilayer optical calculations", arXiv:1603.02720
• Johnson & Christy (1972), PRB 6, 4370 — Ag optical constants
• Mak et al. (2010), PRL 105, 136805 — MoS₂ optical constants
• Bruna & Borini (2009), APL 94, 031901 — Graphene optical constants
• Homola (2003), Anal. Bioanal. Chem. 377, 528 — SPR biosensor review

Authors

Original paper: Md. Mortuza Habib, Ruddro Roy, Md. Mojidul Islam, Mehedi Hassan, Md. Muztahidul Islam, Md. Biplob Hossain