How do you calculate the peak power of a pulsed laser?

This article works step by step through the relationships between average power, pulse repetition rate, pulse energy, pulse duration, peak power and fluence. The characteristics that distinguish a laser source for cleaning are covered in the article on pulse energy and efficiency. Here, the focus is on the calculation.

All numerical examples use assumed values. They do not describe a specific machine; the manufacturer’s specifications are what count.

Which quantities describe a laser pulse

A pulsed laser emits its energy in a regular sequence of short pulses. Five quantities are enough for the calculation:

• Average power in watts (W): the power averaged over many pulses.
• Pulse repetition rate, also called repetition rate, in hertz (Hz) or kilohertz (kHz): the number of pulses per second.
• Pulse energy in joules (J) or millijoules (mJ): the energy content of a single pulse.
• Pulse duration in seconds, for Q-switched lasers often in nanoseconds (ns).
• Peak power in watts or kilowatts (kW): the highest optical power during a pulse.

In addition, there is fluence in joules per square centimetre (J/cm²), the pulse energy relative to the irradiated area.

According to RP Photonics, pulse energy is the time integral of power over the pulse. The quantity most commonly used as pulse duration is the full width at half maximum, i.e. the width of the power curve at half its maximum height, FWHM for short. According to RP Photonics, Q-switched lasers typically generate pulse durations between 100 picoseconds and a few hundred nanoseconds.

Convert units before calculating

Data sheets give millijoules, nanoseconds and kilohertz. The safest approach is to convert to base units before calculating:

• 1 mJ = 0.001 J
• 1 ns = 0.000000001 s, i.e. 10⁻⁹ s
• 1 kHz = 1,000 Hz
• 1 kW = 1,000 W and 1 MW = 1,000,000 W
• 1 mm² = 0.01 cm²

This gives a rule of thumb: millijoules divided by nanoseconds gives megawatts. This is because 0.001 J divided by 0.000000001 s is 1,000,000 W. A pulse with 1 mJ and 100 ns works out at 10 kW — Laserax also gives this example.

From average power to pulse energy

Average power, pulse energy and pulse repetition rate are related as follows:

Average power = pulse energy × pulse repetition rate

Rearranged:

Pulse energy = average power ÷ pulse repetition rate

Example with assumed values: 200 W average power at 20 kHz.

1. Pulse repetition rate in hertz: 20 kHz = 20,000 Hz
2. Pulse energy: 200 W ÷ 20,000 Hz = 0.01 J
3. In millijoules: 0.01 J = 10 mJ

RP Photonics names a precondition: the division is only permissible if no significant energy is emitted between the pulses. If a source additionally emits a weak background, this can contribute noticeably to the average power, and the calculated pulse energy comes out too high.

Peak power from pulse energy and pulse duration

The common approximation, as given for example by Gentec-EO and Laserax:

Peak power ≈ pulse energy ÷ pulse duration

Continued example with 10 mJ and an assumed pulse duration of 100 ns:

1. Pulse energy: 10 mJ = 0.01 J
2. Pulse duration: 100 ns = 0.0000001 s
3. Peak power: 0.01 J ÷ 0.0000001 s = 100,000 W = 100 kW

With the rule of thumb: 10 mJ ÷ 100 ns = 0.1 MW = 100 kW.

Under these assumptions, 200 W of average power therefore becomes pulses with a calculated 100 kW, five hundred times as much. However, peak power does not tell you how concentrated this energy is when it reaches the surface. For that, you need the fluence.

Shortcut via the average power

If the pulse energy is not given, both steps can be combined. Gentec-EO gives:

Peak power ≈ average power ÷ (pulse repetition rate × pulse duration)

The product of pulse repetition rate and pulse duration is the fraction of time taken up by the pulses. At 20 kHz, a pulse follows every 50 microseconds, of which 100 ns is taken up by the pulse itself:

1. Time fraction: 20,000 Hz × 0.0000001 s = 0.002, i.e. 0.2 per cent
2. Peak power: 200 W ÷ 0.002 = 100,000 W = 100 kW

Both routes lead to the same result. The check is still worthwhile: a time fraction of one or more would mean that the pulses follow one another without any gap. In that case, frequency and pulse duration do not match.

Why a higher frequency can mean less energy per pulse

At the same average power, a higher frequency distributes the same energy over more pulses. Example with an assumed 300 W and 100 ns:

• at 30 kHz: 300 W ÷ 30,000 Hz = 10 mJ, peak power 100 kW
• at 60 kHz: 300 W ÷ 60,000 Hz = 5 mJ, peak power 50 kW

Doubling the frequency halves pulse energy and peak power by calculation. This assumes that average power and pulse duration remain the same at both frequencies — with real sources, this cannot simply be taken for granted.

A simulation of a Q-switched fibre laser by RP Photonics shows the background: at the pulse rate of 10 kHz chosen there, the time between two pulses is not sufficient to fully replenish the energy extracted. Anyone calculating with data sheet values should therefore check the frequency for which pulse energy and pulse duration are given.

The rectangular pulse assumption and its limits

The formula pulse energy ÷ pulse duration is exact only for an idealised pulse whose power jumps abruptly to its maximum, stays constant there and ends abruptly. Only for this rectangle is the area under the power curve exactly height times width.

Real pulses rise and fall again. If the full width at half maximum is used, the result deviates from the actual peak power — by how much depends on the pulse shape.

Ophir describes a measurement method that deliberately uses the rectangle assumption: a fast photodiode records the pulse waveform on an oscilloscope. From this, a rectangle is constructed with the same height as the pulse peak and the same area as the curve. The separately measured pulse energy is divided by its width. The oscilloscope shows only the relative waveform.

RP Photonics calls this quantity the effective pulse duration: pulse energy divided by peak power. With it, the rectangle formula is consistent by definition; with the full width at half maximum, it remains an approximation.

Shape factors and real pulse shapes

For known pulse shapes, RP Photonics gives factors, each referring to the full width at half maximum:

• Gaussian-shaped pulse: peak power ≈ 0.94 × pulse energy ÷ pulse duration
• sech²-shaped pulse: peak power ≈ 0.88 × pulse energy ÷ pulse duration

With 10 mJ and 100 ns, this gives around 94 kW or 88 kW respectively instead of 100 kW. For arbitrary pulse shapes, RP Photonics recommends integrating the power over time: the integral must give the pulse energy, and the factor follows from this.

Irregular pulses are more difficult. According to RP Photonics, with heavily distorted pulses a considerable part of the energy can lie in the temporal wings, and the ratio of peak power to pulse energy shifts significantly. Q-switched lasers also often show mode beating, that is, rapid power fluctuations; which peak power is measured then depends on the temporal resolution of the measuring instrument. In the fibre laser simulation mentioned, peaks occur that are shorter than one round trip in the resonator.

A calculated peak power therefore describes an order of magnitude, not a measured value. Values from two sources are only comparable if they are based on the same definition — RP Photonics notes that some authors simply omit the shape factors.

Fluence as energy per area

Fluence describes how concentrated the pulse energy is when it reaches the surface: optical energy per area, usually in J/cm². For a uniformly illuminated flat-top profile, according to Gentec-EO:

Fluence = pulse energy ÷ irradiated area

Example with assumed values: 10 mJ on a round spot with a 1 mm diameter.

1. Radius: 0.5 mm = 0.05 cm
2. Area: π × (0.05 cm)² ≈ 0.00785 cm²
3. Fluence: 0.01 J ÷ 0.00785 cm² ≈ 1.27 J/cm²

Cross-check with the Gentec-EO example: 10 mJ on a flat-top spot with a 4 mm diameter gives 0.0796 J/cm².

Via the average power: fluence = average power ÷ (pulse repetition rate × area). With 200 W and 20 kHz, this also gives around 1.27 J/cm².

The diameter enters as a square: a spot half the size means four times the fluence at the same pulse energy.

Gaussian beam or flat-top profile

Many laser beams become more intense towards the centre. For a Gaussian beam, RP Photonics and Gentec-EO give the fluence on the beam axis as:

Peak fluence = pulse energy ÷ (π × w² ÷ 2)

Here, w is the beam radius at which the intensity has fallen to 1/e², around 13.5 per cent of the maximum. About 86.5 per cent of the power lies within this radius. Compared with the flat-top calculation, the value on the axis doubles: with 10 mJ and a 1 mm 1/e² diameter it is around 2.55 J/cm², and in the Gentec-EO example with 4 mm it is 0.159 J/cm².

Two pitfalls:

• Which diameter? According to RP Photonics, the full width at half maximum of the intensity profile is about 1.18 times the beam radius w and is therefore considerably smaller than the 1/e² diameter. Anyone who inserts it into the 1/e² formula will get a considerably wrong result.
• Forgetting the factor: according to RP Photonics, the factor ½ in the denominator is often overlooked, so that the intensity on the axis is underestimated by a factor of two. For published damage thresholds of optical components, it therefore often remains unclear how the calculation was done.

Gentec-EO recommends measuring the beam profile, for example with a beam camera.

What the calculation does not answer

Peak power and spot area give the peak power density, which Gentec-EO defines as peak power divided by the area of the laser spot. With 100 kW on 0.00785 cm², that is around 12.7 MW/cm² by calculation, and double that on the axis of a Gaussian beam.

Figures like these do not tell you whether a layer will be removed. Three points remain open:

1. Thresholds depend on pulse duration. According to RP Photonics, a damage threshold stated as a fluence is not independent of the pulse duration; the critical value usually rises with longer pulses.
2. Fluence depends on position. It is usually highest on the beam axis and decreases towards the outside. The calculation also applies to a single pulse.
3. The pulse calculation does not fit continuous wave operation. A CW laser does not emit pulses. According to RP Photonics, fluence is only meaningful there together with an exposure duration.

The examples are intended to aid technical understanding. They do not replace the manufacturer’s information on laser class and protective measures, nor a risk assessment.

Calculation steps for a data sheet

If you want to recalculate the characteristics of a pulsed source, you can proceed as follows:

1. Note the average power, pulse repetition rate, pulse energy and pulse duration, and clarify which operating point they apply to.
2. Convert all values into J, s, Hz and cm².
3. Check whether pulse energy × pulse repetition rate roughly gives the average power.
4. Calculate the peak power and clarify whether the pulse duration is given as the full width at half maximum.
5. If the pulse shape is known, apply the shape factor; otherwise read the result as an approximation.
6. Clarify the spot diameter at the workpiece and how it is defined, then calculate the fluence — for the axis of a Gaussian beam with the factor of two.

With the BL Portable Pulse and BL Compact Pulse, Beamlux offers pulsed models with 200 or 300 W at 1064 nm. The pulse calculation does not apply to the BL CW 2000 continuous wave laser; there, power, area and exposure time are what matter. Which pulse parameters belong to a machine should be obtained from the manufacturer for the respective operating point before calculating the peak power of a pulsed laser.

Sources

• RP Photonics Encyclopedia — Peak Power — definition of peak power, shape factors of 0.88 for sech²-shaped and 0.94 for Gaussian-shaped pulses relative to the full width at half maximum, ambiguity due to mode beating in Q-switched lasers.
• RP Photonics Encyclopedia — Pulse Energy — pulse energy as the integral of power over time; calculation from average power and repetition rate and its precondition.
• RP Photonics Encyclopedia — Pulse Duration — full width at half maximum as the most common definition, effective pulse duration, typical pulse durations of Q-switched lasers.
• RP Photonics Encyclopedia — Pulse Repetition Rate — pulse repetition rate as the number of pulses emitted per second.
• RP Photonics Encyclopedia — Fluence — fluence as energy per area, peak fluence of a Gaussian beam, dependence of damage thresholds on pulse duration, fluence in continuous wave operation.
• RP Photonics Encyclopedia — Gaussian Beams — beam radius according to 1/e², power fraction of about 86.5 per cent, full width at half maximum of about 1.18 w, frequently forgotten factor ½.
• RP Photonics — RP Fiber Power, example of pulse generation in a Q-switched fibre laser — simulation with irregular pulse shapes and incomplete replenishment of energy between two pulses.
• Gentec-EO — How to calculate the peak power of a pulsed laser — definition of peak power and peak power density.
• Gentec-EO — laser calculator Peak Power Density and Energy Density — formulas via pulse energy and via average power and repetition rate, factor of two for Gaussian beams, note on their approximate nature.
• Gentec-EO — Quick guide to calculating laser fluence — fluence for flat-top and Gaussian profiles with the worked example of 10 mJ on a 4 mm diameter.
• Ophir — How to measure the peak power of a pulsed laser — measurement with photodiode and oscilloscope, construction of a rectangle of equal area, separate measurement of the pulse energy.
• Laserax — Understanding Laser Powers — peak power as pulse energy divided by pulse duration with the example of 1 mJ at 100 ns, relationship between pulse energy, frequency and power.

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