Two-Photon Absorption Calculator
ChemistryCalculate two-photon absorption rate, absorbed photon flux, and excitation probability from TPA cross-section (GM units), laser intensity, and fluorophore concentration.
10 GM
GM
100 mW
mW
1 μm
μm
1 μM
μM
800 nm
nm
Photon Flux (I)
12,810,000,000,000,000,000,000,000
TPA Absorption Rate
16.41
Irradiance
3,183,098.9
Excitation Probability
0
What is a Two-Photon Absorption?
The Two-Photon Absorption Calculator computes photon flux, irradiance, TPA absorption rate, and excitation probability per pulse for a focused pulsed laser interacting with a TPA-active fluorophore. Enter the TPA cross-section (δ, in GM), laser peak power (mW), beam waist (μm), fluorophore concentration (μM), and wavelength (nm).
Two-photon absorption (TPA) is a nonlinear optical phenomenon where a molecule simultaneously absorbs two photons — requiring the high photon densities available only at the focus of pulsed femtosecond lasers. The TPA rate scales as R = δ × I², where δ is the molecular cross-section in Göppert-Mayer units and I is the photon flux. This quadratic intensity dependence confines excitation to the laser focus, enabling 3D-selective imaging and microfabrication at sub-diffraction volumes.
For the single-photon limit (Beer-Lambert Law for linear absorption), the Beer-Lambert Law Calculator applies A = ε × l × c. For chromophore concentration from UV-Vis absorbance measurements used to verify TPA sample preparation, the Beer-Lambert calculator provides the linear analogue. The Calibration Curve Calculator builds standard curves for quantifying fluorophore concentrations.
How to use this Two-Photon Absorption calculator
- Enter TPA Cross-Section (δ, GM) — from literature for your fluorophore at the laser wavelength. Fluorescein: 37 GM at 800 nm; Rhodamine B: 10–100 GM; Quantum dots: 10,000–50,000 GM.
- Enter Laser Peak Power (mW) — for a Ti:sapphire laser: 10–500 mW typical average power. For pulsed systems, enter peak power = average power / (repetition rate × pulse width).
- Enter Beam Waist (w₀, μm) — the 1/e² radius at the focal point. For a 1.4 NA objective: w₀ ≈ 0.3–0.5 μm.
- Enter Fluorophore Concentration (μM) — for calculating the expected excitation events per volume.
- Read Photon Flux and Absorption Rate — if rate is very low (< 0.01 events/s), the signal will be undetectable; increase power or use a higher-δ dye.
Formula & Methodology
TPA absorption rate:Irradiance: I = P / (π × w₀²) [W/cm²; w₀ in cm] Photon flux: Φ = I / E_photon [photons/cm²/s] E_photon = h × c / λ [J; λ in m] TPA rate: R = δ × Φ² [events/molecule/s] δ in GM = δ × 10⁻⁵⁰ cm⁴·s Excitation probability per pulse (τ_pulse = 100 fs): P_exc = R × τ_pulseWorked example — GFP two-photon imaging in brain tissue: Ti:sapphire laser, 930 nm, 100 mW average, 80 MHz rep rate, 100 fs pulses, focused to w₀ = 0.35 μm (1.0 NA water objective). GFP δ ≈ 6 GM at 930 nm.Peak power = 100 mW / (80 × 10⁶ Hz × 100 × 10⁻¹⁵ s) = 12,500 W = 12.5 kW w₀_cm = 0.35 × 10⁻⁴ cm Irradiance = 12,500 / (π × (0.35×10⁻⁴)²) = 12,500 / 3.85×10⁻⁹ = 3.25×10¹² W/cm² E_photon = (6.626×10⁻³⁴ × 3×10⁸) / (930×10⁻⁹) = 2.14×10⁻¹⁹ J Φ = 3.25×10¹² / 2.14×10⁻¹⁹ = 1.52×10³¹ photons/cm²/s R = 6×10⁻⁵⁰ × (1.52×10³¹)² = 6×10⁻⁵⁰ × 2.31×10⁶² = 1.39×10¹³ events/s → 0.14 events per 10 fsThis rate is adequate for two-photon fluorescence microscopy of GFP-expressing neurons. TIFR's Neuroscience group uses this configuration for imaging mouse cortical neurons expressing channelrhodopsin (for optogenetics) and GFP reporter proteins — part of India's neurotechnology research programme.
Frequently Asked Questions
Two-photon absorption (TPA) is a nonlinear optical process where a molecule simultaneously absorbs two photons to undergo an electronic transition — the combined energy of both photons equals the energy gap (ΔE = 2hν, where ν = laser photon frequency). Unlike single-photon absorption (Beer-Lambert law, linear in intensity), TPA rate is proportional to the square of laser intensity: R_TPA = δ × I², requiring high photon densities available only from pulsed femtosecond or picosecond lasers. TPA was theoretically predicted by Maria Göppert-Mayer (1931) and experimentally confirmed after the laser's invention (1960s).
The Göppert-Mayer unit (GM, named after Maria Göppert-Mayer) is the unit for TPA cross-section (δ): 1 GM = 10⁻⁵⁰ cm⁴·s·photon⁻¹. The TPA cross-section δ measures how efficiently a molecule absorbs two photons. Typical values: Common organic dye molecules: 1–100 GM. Specially designed TPA dyes (for bioimaging): 100–10,000 GM. Quantum dots (CdSe, CdTe): 10,000–50,000 GM. Metal-organic frameworks (MOFs) for TPA: 1,000–100,000 GM. Stilbene derivatives and push-pull π-systems often have large TPA cross-sections. Higher δ → more sensitive TPA detection → better imaging contrast.
Enter TPA Cross-Section (δ, in GM), Laser Peak Power (mW), Beam Waist (w₀, in μm — the 1/e² radius at the focal point), Fluorophore Concentration (μM), and Laser Wavelength (nm). The calculator returns Photon Flux (photons/cm²/s), Irradiance (W/cm²), TPA Absorption Rate (events/molecule/s), and Excitation Probability per pulse (for 100 fs pulses). Default: δ=10 GM, 100 mW, w₀=1 μm, 800 nm wavelength — typical Ti:sapphire laser two-photon microscopy conditions.
TPA requires high peak intensity but can use average powers acceptable for biological samples. Common TPA laser systems: Ti:sapphire laser (700–1050 nm, 100 fs pulses, 80 MHz repetition rate, 100–1000 mW average power) — the gold standard for two-photon microscopy. OPO (Optical Parametric Oscillator) extending range to 1300 nm — for deep tissue imaging. Ytterbium fibre laser (1030–1040 nm) — lower cost, compact. In India: Raman Research Institute (RRI, Bangalore), TIFR (Mumbai), IISc (Bangalore), and JNCASR have ultrafast laser systems for nonlinear optics and two-photon microscopy research, often in collaboration with Indian Institute of Chemical Technology (IICT) for TPA dye synthesis.
Two-photon absorption applications: (1) Two-photon fluorescence microscopy (2PFM): sub-micron resolution imaging of live cells and tissues up to 1 mm deep — advantage over single-photon confocal microscopy is reduced phototoxicity and better tissue penetration. (2) 3D microfabrication: two-photon polymerisation (2PP) achieves sub-100 nm resolution by photo-polymerising only at the laser focus — used in microfluidics, photonic crystals, MEMS. (3) Photodynamic therapy (PDT): TPA sensitisers with large cross-sections for deep tumour treatment — avoiding superficial light absorption. (4) Optical data storage: 3D data recording using TPA-active materials. (5) Two-photon optogenetics: selective neural activation in deep brain tissue. Indian research groups at TIFR, NCL, and JNCASR are active in all these areas.
For a Gaussian beam focused to a beam waist w₀ (1/e² radius): Irradiance at focus = I = P / (π × w₀²), where P = peak power (W), w₀ in cm. Photon flux = I / E_photon = I × λ / (h × c), where λ is in m, h = 6.626×10⁻³⁴ J·s, c = 3×10⁸ m/s. Example: 100 mW average power, 100 fs pulses, 80 MHz repetition rate, focused to w₀ = 1 μm. Peak power ≈ 100 mW × 1/(80 MHz × 100 fs) = 100 mW / 8 × 10⁻³ = 12.5 kW peak. For TPA, it is the PEAK power that drives the quadratic process, not the average power.
One-photon (confocal) microscopy: excite at λ/2 (e.g., 488 nm for GFP); excitation occurs along the entire laser path through the sample — causes photobleaching and phototoxicity outside the focal plane. Two-photon microscopy: excite at λ (e.g., 930 nm for GFP, using TPA); excitation occurs ONLY at the focal point (where photon flux is high enough for TPA quadratic process) — inherently 3D-selective, no out-of-focus bleaching. Additional advantage: NIR wavelengths (700–1000 nm) penetrate biological tissue more deeply (less scattering and absorption) than UV-Vis — can image 500–1000 μm into intact brain slices. This is why two-photon microscopy is used at TIFR's neuroscience division for in-vivo brain imaging in mice.
Common TPA fluorophores in bioimaging: Rhodamine B (δ ≈ 10 GM at 840 nm): classic dye, moderate TPA. FITC/fluorescein (δ ≈ 37 GM at 780 nm): cell labelling. Calcein (δ ≈ 40 GM): intracellular Ca²⁺ imaging. BODIPY dyes (δ = 20–200 GM): lipid membrane and organelle labelling. Genetically encoded fluorescent proteins: GFP (δ ≈ 6 GM), mCherry, mCerulean. Quantum dots (δ ≈ 50,000 GM): extremely bright TPA probes; CdSe/ZnS core-shell nanocrystals — being explored at CSIR-NCL Pune and JNCASR. Near-IR dyes: cyanine dyes (Cy5, Cy7) and aggregation-induced emission (AIE) materials with δ > 1000 GM for deep tissue imaging.
Two-photon polymerisation (2PP / direct laser writing) uses TPA to locally polymerise photoinitiator-containing resins at the laser focus: only the voxel (3D pixel) at the focus reaches the threshold dose to polymerise — enabling truly 3D structures with sub-micron resolution (100–200 nm features). Applications: microlenses, photonic waveguides, biomedical scaffolds, microfluidics, MEMS. Indian institutions active in 2PP: IIT Bombay (femtosecond laser microfabrication lab), IISc Bangalore (Centre for Nanoscience), RRCAT Indore (Indus-2 synchrotron — also runs ultrafast laser labs), TIFR (femtosecond spectroscopy group). Commercial 2PP systems (Nanoscribe, UpNano) enable < 200 nm feature fabrication — not yet widely available in India but growing with AIC-IITB and BIRAC-supported deep-tech startups.
Measurement methods for δ: (1) Two-photon excited fluorescence (TPEF): compare two-photon fluorescence from the sample to a reference dye with known δ (e.g., fluorescein in water, δ = 37 GM at 800 nm). δ_sample = δ_ref × (F_sample / F_ref) × (φ_ref / φ_sample) × (c_ref / c_sample) × (η_ref² / η_sample²), where F = TPEF signal, φ = fluorescence quantum yield, c = concentration, η = refractive index. (2) Z-scan technique: measures non-linear absorption by scanning a sample through the laser focus. (3) Pump-probe spectroscopy: time-resolved measurement of excited state population. Most δ measurements worldwide are done by TPEF relative to fluorescein — the reference standard used by Drobizhev, Makarov, and Webb's comprehensive tables.