Two-Photon Absorption Calculator
Calculate the two-photon absorption excitation rate, cross-section, and related parameters for nonlinear optical processes.
What is Two-Photon Absorption?
Two-photon absorption (TPA) is a third-order nonlinear optical process in which a molecule or material simultaneously absorbs two photons to transition from a lower energy state to a higher energy state. Unlike conventional (one-photon) absorption, which depends linearly on light intensity, TPA scales with the square of the intensity, making it inherently a nonlinear phenomenon.
The concept was first predicted theoretically by Maria Goeppert-Mayer in her 1931 doctoral dissertation, long before lasers existed to demonstrate the effect experimentally. The unit of two-photon absorption cross-section, the Goeppert-Mayer (GM), is named in her honor. One GM equals 10-50 cm4·s/photon.
Because TPA requires high photon densities, the process only became experimentally accessible after the invention of the laser in 1960. Today, TPA is widely used in fluorescence microscopy, 3D microfabrication, photodynamic therapy, and optical data storage, among other applications.
How Does Two-Photon Absorption Work?
In two-photon absorption, two photons arrive at a molecule nearly simultaneously (within about 10-16 seconds of each other). Each photon individually does not have enough energy to excite the molecule, but their combined energy matches the energy gap between the ground state and an excited electronic state.
The process involves a transient intermediate known as a virtual state. Unlike real quantum mechanical states, the virtual state does not correspond to any eigenstate of the molecule. Instead, it represents a very short-lived superposition that exists only during the interaction with the photons. The lifetime of this virtual state is governed by the time-energy uncertainty principle and is on the order of a few femtoseconds.
The energy diagram is straightforward: a molecule in the ground state S0 absorbs the first photon and is promoted to the virtual state. Almost instantaneously, it absorbs a second photon and reaches the real excited state S1 (or a higher state). Mathematically, the total energy absorbed is:
Because both photons typically come from the same laser, they have identical wavelength. This means the excitation wavelength for TPA is roughly twice that used for one-photon excitation of the same transition. For example, a fluorophore that absorbs one-photon light at 400 nm can be excited via two-photon absorption at approximately 800 nm.
Two-Photon Absorption Cross-Section
The two-photon absorption cross-section, denoted σ2, quantifies how efficiently a molecule undergoes two-photon absorption. It has units of cm4·s/photon, and by convention is expressed in Goeppert-Mayer units (GM):
The cross-section depends on the molecular structure, specifically the electronic transition dipole moments and the energy difference between states. Molecules with extended conjugation, strong donor-acceptor character, or planar aromatic structures tend to have large σ2 values.
| Molecule / Material | σ2 (GM) | Wavelength (nm) |
|---|---|---|
| Fluorescein | ~38 | 780 |
| Rhodamine B | ~150 | 840 |
| EGFP (Enhanced Green Fluorescent Protein) | ~40 | 920 |
| Coumarin 307 | ~15 | 776 |
| Stilbene derivatives | 100 - 1000 | 700 - 900 |
| Quantum dots (CdSe) | 10,000 - 50,000 | 800 |
Two-Photon Excitation Rate Formula
The rate of two-photon excitation for a single molecule is given by:
Where:
- RTPA is the two-photon excitation rate (photons absorbed per second per molecule)
- σ2 is the two-photon absorption cross-section (cm4·s/photon)
- I is the instantaneous laser intensity (W/cm2)
- ℏω is the energy of a single photon (J)
The key feature of this equation is the quadratic dependence on intensity. This means that TPA occurs preferentially at the focal point of a tightly focused laser beam, where the intensity is highest. Outside the focal volume, the intensity drops off rapidly and TPA is negligible. This inherent optical sectioning capability is the foundation of two-photon microscopy.
To derive this, consider that the photon flux (number of photons per unit time per unit area) is Φ = I / (ℏω). Two-photon absorption requires two photons, so the rate depends on the probability of two photons being present at the molecule simultaneously, which scales as Φ2. Multiplying by the cross-section σ2 gives the absorption rate.
Pulsed vs. Continuous Wave Lasers
Although TPA can in principle be achieved with continuous-wave (CW) lasers, pulsed lasers are overwhelmingly preferred in practice. The reason is the quadratic intensity dependence: concentrating the same average power into short pulses dramatically increases the peak intensity, and therefore the TPA rate.
For a pulsed laser, the peak intensity is related to the average power by:
Where:
- Pavg is the time-averaged laser power
- f is the repetition rate (number of pulses per second)
- τ is the pulse duration (width of each pulse)
- A is the focal area at the sample, A = π w02
Consider a typical Ti:Sapphire oscillator delivering 10 mW average power at 80 MHz repetition rate with 100 fs pulse duration focused to a 0.5 μm beam waist. The peak intensity is on the order of 1011 W/cm2, whereas a CW laser with the same average power would produce only about 106 W/cm2. The TPA rate with the pulsed laser is therefore approximately 1010 times higher.
This enormous enhancement explains why mode-locked femtosecond lasers, particularly Ti:Sapphire oscillators operating near 800 nm, became the standard source for two-photon microscopy.
Measurement Techniques
Measuring two-photon absorption cross-sections accurately is challenging because the effect is weak and requires high intensities. Several experimental methods have been developed:
- Z-scan technique: The sample is translated along the beam axis (z-direction) through the focus of a tightly focused laser beam. By measuring the transmitted intensity as a function of sample position, both the nonlinear refractive index and the TPA coefficient can be extracted. The open-aperture Z-scan isolates the absorption contribution.
- Two-photon excited fluorescence (TPEF): The sample is excited with a pulsed laser, and the resulting fluorescence is measured. By comparing the fluorescence intensity to a reference standard with a known σ2 (such as fluorescein or rhodamine B), the unknown cross-section can be determined. This method requires knowledge of the fluorescence quantum yield.
- Nonlinear transmission: The transmission of a laser beam through the sample is measured as a function of input intensity. A decrease in transmission at high intensities indicates two-photon absorption. The TPA coefficient β can be extracted by fitting the data to the appropriate propagation equation.
- Photoacoustic methods: The energy absorbed via TPA is converted to heat, which generates an acoustic wave that can be detected with a microphone or piezoelectric transducer.
- White-light continuum pump-probe: A broadband probe pulse measures the absorption spectrum changes induced by a pump pulse, providing spectral information about TPA across a wide wavelength range.
Applications of Two-Photon Absorption
Two-photon absorption has enabled a wide range of applications across science, medicine, and technology:
Two-photon fluorescence microscopy: Perhaps the most impactful application, two-photon microscopy allows deep tissue imaging with subcellular resolution. Because the excitation light is in the near-infrared (typically 700-1000 nm), it penetrates tissue more deeply than visible light. The inherent optical sectioning from the quadratic intensity dependence eliminates the need for a confocal pinhole, improving signal collection. Two-photon microscopy is now a workhorse technique in neuroscience, developmental biology, and immunology.
3D microfabrication and two-photon polymerization: TPA can initiate polymerization reactions in a photoresist material with sub-diffraction-limit resolution. By scanning the laser focus in three dimensions, complex 3D microstructures can be fabricated. This technique achieves feature sizes below 100 nm and is used to create micro-optical elements, microfluidic devices, and metamaterials.
Photodynamic therapy (PDT): In PDT, a photosensitizer molecule is activated by light to generate reactive oxygen species that destroy cancer cells. Using TPA for excitation allows the use of near-infrared light, which penetrates deeper into tissue than the visible light required for one-photon excitation. This extends the treatment depth and reduces damage to surrounding healthy tissue.
Optical data storage: TPA enables writing and reading data in three dimensions within a photorefractive or photochromic medium. Because the absorption is confined to the focal volume, data can be stored in multiple layers within a single disk, potentially increasing storage density by orders of magnitude compared to conventional optical media.
Optical limiting: Materials with strong TPA can serve as optical limiters that protect sensitive detectors and eyes from intense laser radiation. At low intensities the material is transparent, but at high intensities TPA increases the absorption, clamping the transmitted intensity to a safe level.
Free Electron Absorption
An important fundamental question is why a free electron cannot absorb a single photon. The answer lies in the simultaneous conservation of energy and momentum.
For a free electron absorbing a photon, the conservation laws require:
Momentum: pe + ℏk = pe'
However, these two equations cannot be satisfied simultaneously for a free electron. The photon carries very little momentum (ℏk = ℏω/c) relative to its energy. If we consider an electron initially at rest absorbing a photon, the final electron would need to have both the photon's energy and its momentum. But for a massive particle, E = p2/(2m), which gives a different energy-momentum relationship than that of the photon (E = pc). The two constraints are incompatible.
Absorption can occur when a third body is present to absorb the excess momentum. In bound systems (atoms, molecules, solids), the nucleus or crystal lattice provides this momentum sink. This is why photon absorption readily occurs in atoms and molecules but not for truly free electrons in vacuum.
In the context of two-photon absorption, the molecule itself serves as the system that simultaneously conserves energy and momentum. The two photons can have their momenta arranged so that the total momentum transfer is accommodated by the molecular recoil, though in practice the recoil energy is negligibly small compared to the electronic transition energy.