Two-Photon Absorption Coefficient Calculator

Two-Photon Absorption Calculator

A two-photon absorption calculator turns three laser settings (wavelength, intensity, and pulse duration) into a single illustrative figure, labelled β and reported in cm/GW. Two-photon absorption is a process in which a molecule absorbs two photons of light at almost the same instant instead of one, and it only happens at high light intensity. The figure this page produces is a teaching aid for the arithmetic, not a measured or predicted coefficient for any real substance; the section below explains why.

What Is Two-Photon Absorption?

In ordinary, single-photon absorption, a molecule absorbs one photon of light and jumps to a higher energy state. Two-photon absorption (TPA) is different: the molecule absorbs two photons within femtoseconds of each other and uses their combined energy to make the same jump. A femtosecond is one millionth of one billionth of a second.

This double absorption is rare. It only happens often enough to matter when light is extremely intense, because two photons must arrive almost simultaneously at the same molecule. Ordinary light, even sunlight, is far too weak. Lasers that pack their energy into short, intense pulses make TPA practical to observe and use.

The physicist Maria Göppert-Mayer predicted two-photon absorption in 1931, as part of her doctoral thesis. No light source of the time was intense enough to show it. The laser, invented in 1960, changed that. In 1961, researchers Wilhelm Kaiser and C. G. B. Garrett at Bell Labs observed two-photon absorption for the first time, using a ruby laser and a crystal doped with europium. Göppert-Mayer later shared the 1963 Nobel Prize in Physics for other work on nuclear structure.

Two-Photon Absorption Coefficient Formula

The calculator applies one fixed formula:

$\beta = K \times \frac{I \times \tau}{\lambda^2}$

Where:

• β — two-photon absorption coefficient, in cm/GW
• K — a fixed constant, set to 1.5 in this calculator
• I — intensity of the light, converted from W/cm² to GW/cm² (divide by 10⁹)
• τ — pulse duration, in femtoseconds (fs)
• λ — wavelength of the light, in nanometers (nm)

Why this formula is a placeholder, not physics

The calculator states this openly, and so does this article. In real optics, β is an intrinsic property of the material. It appears in the equation dI/dz = −βI², the two-photon analogue of ordinary Beer-Lambert absorption, and it depends only on the material's band gap, its linear refractive index, its Kane energy, and the photon energy. Beam intensity and pulse duration do not appear in it. Doubling the laser power does not double a material's β.

The formula above does make β rise with intensity and pulse duration, so it cannot be describing that physical quantity. Its units do not reduce to cm/GW either: GW/cm² × fs ÷ nm² is not a length divided by a power. The number is therefore an illustration of the arithmetic, and the calculator shows a standing warning saying so.

Getting a real β needs either a direct measurement, usually by the Z-scan method, or a material model such as the two-band scaling rule of Sheik-Bahae, Hagan, and Van Stryland, β = K·√E_p·F₂(2ħω/E_g) / (n₀²·E_g³). Here E_g is the band gap, n₀ the linear refractive index, E_p the Kane energy, and F₂ a fixed function of the ratio 2ħω/E_g. That model takes material parameters this calculator does not ask for.

How Each Variable Behaves

Wavelength (λ) appears squared in the denominator, so the figure falls as wavelength increases, following an inverse-square relationship. Going from 800 nm to 1000 nm, with intensity and pulse duration unchanged, multiplies it by (800/1000)² = 0.64, a drop of 36% rather than a drop of more than half. Most two-photon experiments use near-infrared light, roughly 700–1000 nm, because it penetrates biological tissue well. Titanium-sapphire (Ti:Sapphire) lasers, tunable around 800 nm, are the most common light source for this reason.

Intensity (I) must reach roughly 10¹⁰ to 10¹⁴ W/cm² for two-photon absorption to become significant in the laboratory, and the calculator flags values outside that band. Sunlight at the earth's surface is only about 0.1 W/cm², far too weak. Reaching these intensities requires pulsed lasers, which concentrate their energy into brief bursts rather than spreading it out continuously.

Pulse duration (τ) typically ranges from 10 to 1000 femtoseconds in TPA experiments. Shorter pulses concentrate a given amount of energy into less time, producing higher peak intensity, but they also require more complex and costly laser equipment.

Worked example

Using the formula with typical Ti:Sapphire laser parameters:

• Wavelength: 800 nm
• Intensity: 10¹² W/cm²
• Pulse duration: 100 fs
• K = 1.5

First, convert intensity to GW/cm²: 10¹² ÷ 10⁹ = 1000 GW/cm².

Then apply the formula:

β = 1.5 × (1000 × 100) ÷ 800² = 150,000 ÷ 640,000 = 0.234375, which the calculator displays as 0.2344 cm/GW

Keeping intensity and pulse duration the same but changing the wavelength to 1000 nm gives:

β = 1.5 × (1000 × 100) ÷ 1000² = 150,000 ÷ 1,000,000 = 0.15, displayed as 0.1500 cm/GW

That is a 36% decrease, matching the inverse-square relationship described above.

What the Calculator Shows

The page reports one number, the coefficient, in cm/GW. Values between 0.001 and 999 are shown to four decimal places; anything smaller or larger switches to scientific form, such as 1.5000 × 10^-5. A panel underneath repeats the arithmetic step by step with the converted intensity, so the result can be checked by hand.

A warning box sits above the inputs at all times, stating that the number is not a validated physical coefficient. Each input is checked separately. A value of zero or less is rejected with an error, and a positive value outside the usual experimental band raises a range warning but still produces a result. A diagram beside the result shows the beam entering and leaving a sample, with the current wavelength, intensity, and coefficient labelled.

Applications of Two-Photon Absorption

Two-photon microscopy. Because absorption only happens where the laser beam is tightly focused, this technique excites fluorescent molecules only at a single point deep inside a sample. It is widely used to image living brain tissue, since it avoids damaging or bleaching regions outside the focal point. Winfried Denk, James Strickler, and Watt Webb built the first two-photon microscope at Cornell University in 1990.

Photodynamic therapy. Near-infrared light reaches deeper into tissue than visible light. Two-photon excitation can activate a light-sensitive drug only at the laser's focal point, which helps target tumors while sparing surrounding healthy tissue.

3D optical data storage. Focusing a laser at different depths inside a block of material lets two-photon absorption mark data at that depth alone, offering higher storage density than surface-based formats like DVDs.

Two-photon microfabrication. Also called two-photon polymerization, this technique hardens light-sensitive resin only at the laser's focal point, producing 3D-printed structures with features smaller than 100 nanometers.

Optical limiting. Materials with a high β stay transparent at normal light levels but absorb strongly under an intense laser pulse. This property is used in protective eyewear and sensor filters against accidental laser exposure.

How to Calculate the Absorption Coefficient Manually

1. Convert the light's intensity from W/cm² to GW/cm² by dividing by 10⁹.
2. Multiply the result by the pulse duration in femtoseconds.
3. Multiply by the constant K (1.5 in this simplified model).
4. Divide by the wavelength in nanometers, squared.

The calculator performs these same four steps and flags any input that falls outside the typical experimental range: 400–1200 nm for wavelength, 10¹⁰–10¹⁴ W/cm² for intensity, and 10–1000 fs for pulse duration. Values outside these ranges still produce a number.

Frequently asked questions

What is two-photon absorption? It is a process where a molecule absorbs two photons of light almost simultaneously, instead of one, to reach a higher energy state. It only becomes significant at very high light intensity.

What does the coefficient β measure? In optics, β measures how strongly a material undergoes two-photon absorption, in units of cm/GW. A higher β means the material absorbs more strongly at a given intensity. Published values span a wide range: fused silica is around 10⁻⁶ cm/GW near 800 nm, while gallium arsenide is roughly 26 cm/GW at 1064 nm. The figure this page prints is not one of those measured values.

Why does two-photon absorption need such intense light? Two photons must strike a molecule within femtoseconds of each other, which is very unlikely at ordinary light levels. Pulsed lasers concentrate energy into short bursts to make this likely enough to observe and use.

How does wavelength affect the result in this calculator? The result falls as the square of the wavelength increases. Moving from 800 nm to 1000 nm, with other settings unchanged, reduces it by 36%, from 0.234375 to 0.15 in the worked example above.

Can this formula be used for three-photon absorption? No. Three-photon absorption is described by its own coefficient, and the absorbed power grows with the cube of the intensity rather than its square. It needs a separate model.

How accurate is the simplified formula? It is not accurate, and it is not meant to be. It has no material properties in it, and its units do not reduce to cm/GW, so its output should not be quoted as a coefficient for any substance. Use a measured value, or a material model such as the Sheik-Bahae two-band expression, for real work.

Does a real coefficient change when the laser is turned up? No. β is a property of the material and the wavelength, so it stays the same as intensity and pulse duration change. What changes is how much light the material absorbs, because two-photon absorption removes energy in proportion to the square of the intensity.

References

1. Göppert-Mayer, M. (1931). "Über Elementarakte mit zwei Quantensprüngen." Annalen der Physik, 401(3), 273–294.
2. Kaiser, W., & Garrett, C. G. B. (1961). "Two-Photon Excitation in CaF₂: Eu²⁺." Physical Review Letters, 7(6), 229–231.
3. Denk, W., Strickler, J. H., & Webb, W. W. (1990). "Two-photon laser scanning fluorescence microscopy." Science, 248(4951), 73–76.
4. Rumi, M., & Perry, J. W. (2010). "Two-photon absorption: an overview of measurements and principles." Advances in Optics and Photonics, 2(4), 451–518.
5. Sheik-Bahae, M., Hagan, D. J., & Van Stryland, E. W. (1990). "Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption." Physical Review Letters, 65(1), 96–99.
6. Sheik-Bahae, M., Hutchings, D. C., Hagan, D. J., & Van Stryland, E. W. (1991). "Dispersion of bound electronic nonlinear refraction in solids." IEEE Journal of Quantum Electronics, 27(6), 1296–1309.