Tool · Laser characterization

Pulsed Laser Calculator

How much energy is in each pulse, how high does the power peak, and what fluence reaches the target? From a laser’s average power, repetition rate, pulse duration and spot size, the calculator gives the pulse energy, peak power, duty cycle, photons per pulse, and the peak fluence and intensity at the focus. Background: Q-switching, mode-locking, fluence, and time-bandwidth product.

Laser

Average power P (W)

Repetition rate f

Pulse duration τ (FWHM)

Pulse shape

Wavelength λ (nm)

Spot diameter, 1/e² (µm)

Q-switched and gain-switched lasers are usually close to Gaussian in time; passively mode-locked lasers are close to sech². A flat-top describes a modulated diode or a gated source. For the spot size of a focused beam, see the Gaussian beam calculator and Gaussian beam.

Presets

The presets are round examples of each kind of source, not the specifications of particular products.

Readouts

One pulse, three shapes at the same energy and width

Gaussiansech²flat-topselected shape

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The closed forms are compared with values worked out by hand, each pulse shape is integrated numerically to confirm its peak-power factor, and the Gaussian spot’s fluence is integrated over the plane to confirm that it returns the pulse energy.

Check	Expected	Computed	Tolerance

The expected values follow the pulse and beam relations in Siegman, Lasers, and Weiner, Ultrafast Optics, evaluated by hand for the stated cases. The tolerance is the largest relative difference from Expected that still passes.

The model

A train of pulses with average power $P$ at repetition rate $f$ carries a pulse energy and peak power

E = \frac{P}{f}, P_{p k} = k \frac{E}{τ}

where $τ$ is the full width at half maximum of the pulse power and $k$ depends on its shape: 1 for a flat-top pulse, $2 \sqrt{\ln 2 / π} = 0.9394$ for a Gaussian and $\ln (1 + \sqrt{2}) = 0.8814$ for a sech² pulse. The duty cycle is $τ f$ . Focused to a spot with a Gaussian profile of 1/e² radius $w$ , the fluence and intensity on the axis are

F_{0} = \frac{2 E}{π w^{2}}, I_{0} = \frac{2 P_{p k}}{π w^{2}}

twice the values averaged over the 1/e² area. The number of photons in a pulse is $E λ / (h c)$ . The model takes every pulse as identical and isolated; it does not include the pedestal or satellite pulses of a real amplifier, which carry energy without adding to the peak.

A pulse cannot have a narrower spectrum than its duration allows. An unchirped pulse has the smallest product of the FWHM bandwidth and the duration, the time-bandwidth product $Δ ν τ = K$ , with $K = 2 \ln 2 / π = 0.4413$ for a Gaussian, 0.3148 for a sech² and 0.8859 for a flat-top pulse; in wavelength $Δ λ = λ^{2} Δ ν / c$ . A 100 fs sech² pulse at 800 nm therefore spans at least 3.148 THz, or 6.721 nm. A chirped pulse has more bandwidth than this for its duration.

Worked example

A Q-switched laser at 1064 nm with 1 W of average power at 1 kHz delivers 1 mJ per pulse. In 10 ns Gaussian pulses that is a peak power of 93.94 kW and a duty cycle of 10⁻⁵. Focused to a 100 µm spot, the peak fluence is 25.46 J/cm² and the peak intensity 2.392 × 10⁹ W/cm²; each pulse holds 5.356 × 10¹⁵ photons.

References: A. E. Siegman, Lasers (University Science Books, 1986). A. M. Weiner, Ultrafast Optics (Wiley, 2009). O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010).