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Use of radiation pressure for measurement of high-power laser emission

Paul A. Williams, 1,* Joshua A. Hadler, 1 Robert Lee, 2 Frank C. Maring, 2 and John H. Lehman 1

1National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA2Scientech Inc., 5649 Arapahoe Ave., Boulder, Colorado 80303, USA

*Corresponding author: paul.williams@nist.gov

Received June 27, 2013; revised August 20, 2013; accepted September 4, 2013;posted September 16, 2013 (Doc. ID 192991); published October 15, 2013

We demonstrate a paradigm in absolute laser radiometry where a laser beam ’ s power can be measured from itsradiation pressure. Using an off-the-shelf high-accuracy mass scale, a 530 W Yb-doped fiber laser, and a 92 kWCO 2 laser, we present preliminary results of absolute optical power measurements with inaccuracies of better than7% to 13%. We find negligible contributionfrom radiometric (thermal) forces. We also identify this scale ’ s dynamic-force noise floor for a 0.1 Hz modulation frequency as 4 μ N⁄ Hz 1 ⁄ 2 or, as optical power sensitivity, 600 W ⁄ Hz 1 ⁄ 2 .

OCIS codes: (040.0040) Detectors; (040.6808) Thermal (uncooled) IR detectors, arrays and imaging; (350.4800)Optical standards and testing.

http://dx.doi.org/10.1364/OL.38.004248

In the 50 years since the invention of the laser, fundamen-tally accurate measurements of laser output power have

been calorimetric in nature. That is, a sensor head max-imally absorbs the laser light, and the laser energy or power is measured from the change in temperature of thetarget. This technique affords impressive 0.005% to 3% absolute accuracies for various power levels [ 1– 4] but be-comes increasingly difficult as laser powers increaseabove roughly 10 kW. We describe here a different para-digm in which the laser power is measured not by absorb-ing it, but rather by reflecting it and measuring the totalradiation force experienced by the mirror. We show thatfor power levels above a few hundred watts, this meas-urement can be carried out with a portable, off-the-shelf mass-weighing scale. This presents the possibility of low-cost, accurate, power-scalable, and fast characterizationof high laser powers.

In the most general sense, accurate optical power mea-surements require a comparison between optical power and a separate fundamental quantity. In traditional laser calorimeters, this quantity is electrical energy. An opti-cally absorbing thermal mass is injected with laser light,and that optical energy is measured by comparing thecalorimeter ’s temperature increase to that induced byan equivalent injection of electrical energy. With this kindof thermal design, the size and response period of a calo-rimeter will scale linearly with its optical energy capac-ity. For example, calorimeters capable of measuring upto hundreds of kilojoules of optical energy have volumeson the order of 1 m3 and response periods of tens of minutes and even longer recovery (cooling) periods. An improvement to this is a flowing-water power meter where laser light is injected into an absorbing cavity, andthe heat is removed by rapidly flowing water. The optical power is measured from the water ’s flow rate and tem- perature change [ 4]. This gives an improved responsetime (tens of seconds) that is less dependent on optical power capacity, but the size (cubic meters) still scaleslinearly with power.

Measuring optical power by use of radiation force, onthe other hand, is dramatically simpler because it elimi-nates the requirement to absorb the laser power, making

the basic design at least an order of magnitude smaller (determined only by the laser-beam diameter, rather than

the optical power). Absolute optical power can be mea-sured fundamentally by comparison to a reference forceor equivalent mass. Such reflective power measurementsare nonperturbative in the sense that laser power can beassessed in real time while the fully reflected light isstill available for use. The slow process of heating andcooling massive calorimeters is no longer required,enabling fast, power-independent response periods withno recovery period.

The concept of radiation pressure has long been under-stood and demonstrated [ 5– 7]. Photons carry momen-tum, so a laser beam reflecting from a mirror impartsa radiation force F due to the change in photon momen-tum, which is proportional to the optical power P :

F 2 P ⁄ c r cos θ ; (1)

where r R 1 − R α ⁄ 2 accounts for the fact that anabsorbed photon imparts all its momentum, and a re-flected photon imparts twice its momentum. R is the mir-ror reflectivity, α indicates the fraction of nonreflectedlight absorbed by the mirror, θ is the angle of incidence,and c is the speed of light. Equation ( 1) describes a maximum power-to-force conversion factor of 2 ⁄ c 6 .67 × 10 −9 N⁄ W for normal incidence on a perfectly re-flecting mirror.

The idea of radiation pressure as a means of measuringoptical energy has been previously proposed [ 8– 12].However, all of these previous designs were based ona torsion balance, which has several limitations. Theseinclude a slow time response due to long oscillation periods (tens to hundreds of seconds [ 13]), difficulty inscaling to larger beam diameters, operation in a vacuumenvironment, and inability for fundamental calibration toa force standard. A recent work [ 14] achieves impressiveresults at milliwatt power levels using a disc-pendulumapproach but lacks portability and power scalability.Today, with industrial, defense, and research laser power levels from kilowatt to megawatt, and robust commercialscales with precisions of 1 nN (0.1 μ g), a radiation-

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pressure approach to portable absolute power measure-ment in high-power lasers becomes practical.

Our goal in this work is to demonstrate that radiation pressure can be measured for kilowatt-level laser powerswith existing scale technology. Our experimental setupis shown in Fig. 1. The scale was an off-the-shelf direct-loading force-restoration balance [ 15] with a 100 nN(10 μ g) resolution. Our 530 W Yb-doped fiber laser source, centered at 1071 nm with a 5 nm full width-at-

half-maximum bandwidth is capable of providing 3.5 μ Nof radiation force. This is at the lower end of our scale ’soperation range, but provides sufficient signal-to-noiseratio to demonstrate feasibility. The weakly focused lightimpinges on a first-surface dielectric mirror (25 mmdiameter) mounted on the shaft of the scale. The angleof incidence θ of the laser beam with respect tothe mirror normal was adjustable between 7° and 45°.We measured the mirror reflectance to be R 0 .997 over this angular range at the operating wavelength.

The direct-loading balance design is integral to the practical operation of this radiation-pressure power meter. The direct-load mechanism [ 15] does not require

gravity for its mechanical operation, so, by removing thespring that compensates for the weight of the balance pan, we were able to operate the balance in an orienta-tion where the force is measured horizontally. Thissignificant feature allows the laser beams to travel andbe measured in a horizontal direction (parallel to the plane of the floor). This avoids vertically propagatinglight, which is a practical laser safety issue and particu-larly important at the highest laser powers.

To isolate the radiation force from noise effects due totemperature drift and ambient air current, the amplitudeof the laser was current-modulated (100% depth) witha 0.1 Hz square wave. The scale ’s internal electronicsinclude a digitizer circuit with an effective response period of ∼2 to 4 s. The modulated laser beam irradiatedthe mirror for 5 to 10 min. for signal averaging, while thedigitized scale reading was acquired at a sample rate of 18 Hz and converted from mass to force by use of thegravitational constant g 9 .80 m ⁄ s 2 . Typical raw outputis shown in Fig. 2, displaying a drift due to thermal effectsand random air currents. We extracted (in software) theamplitude of the modulated radiation force F opt fromthe drifting background by use of a quadrature hetero-dyne approach:

F opt gπ

2 h m t sin ω t i2 h m t cos ω t i2q ; (2)

where m t is the time-dependent mass reported by thescale, ω is the angular modulation frequency of the laser,and brackets indicate a time average over the laser injec-tion interval. The force on the scale was measured as a function of laser power from 0 to ∼530 W. Results areshown as solid circles in Fig. 3, where the horizontalaxis is effective optical power Pr cos θ , and the solidline indicates the theoretical prediction of Eq. ( 1). As anestimate of the noise, with no laser power incident on

the scale, we measure 0.1 μ N of force. Within this noise,we see a linear response to laser power with a best-fitslope of 6 .23 × 10 −9 N ⁄ W (∼7% below the theoretical2 ⁄ c value).

To determine the upper power range, we operationallytested our meter with a 100 kW cw CO 2 laser. In order toaccommodate the larger 10 cm beam diameter withoutdrastically increasing the mirror mass, we used a 20 cmdiameter silicon wafer coated with gold and a surfacedielectric layer, providing R 0 .998 at 10.6 μ m and 45°incidence angle. With the increased mirror area, the bal-ance was particularly susceptible to slight air currents,and so it was protected by a windowless housing withcylindrical baffles surrounding the input and output light paths. We measured the resultant force on the scale for a 3 s exposure of unmodulated 24 and 92 kW laser powersat a nominal 45° angle of incidence. The latter is shown inFig. 4. We experienced no damage during the exposure,but did see a delayed drift in the scale background forcereading, which might be due to a temperature rise in thescale mechanical system due to residual heating effectsfrom the light. The measured force was 13% lower thanexpected, and the results are shown as squares in Fig. 3.

We evaluate the significance of the discrepancies inthe measured force-to-power ratio by considering themeasurement uncertainty. For the 1071 nm fiber laser

Fig. 1. Layout of radiation pressure measurement. In thiswork, the laser beam propagates entirely in a horizontal plane,requiring the scale to operate in a “ vertical ” orientation with themirror surface in a vertical plane.

6

4

2

0 S c a l e

f o r c e

( N )

5004003002001000

Time (s)

Fig. 2. Scale reading (in force units) for 530 W incident power modulated at 0.1 Hz. The average y -axis value is arbitrary sinceit includes the initial (nonzero) mass reading of the scale.

0.1

1

10

100

M e a s u r e d

f o r c e

( N )

101

102

103

104

105

Effective power (W)

Fig. 3. Measured radiation force versus effective laser power (includes mirror reflectivity and angle of incidence). Circles are1071 nm laser; squares are 10.6 μ m laser. The solid line indicatesthe theoretical force-to-power slope.

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measurements, the incident laser power was measuredindirectly and is known to only ∼2% uncertainty. An ad- vantage of a true mass-reading scale (in contrast to a tor-sion balance) is that the force reading can be calibratedwith a simple mass artifact. Our scale was calibrated witha known mass of 20 g (0.196 N), and its accuracy near theoperation range was tested by comparison with a set of calibrated masses down to 1 mg (10 μ N), see Fig. 5. But

the 530 W fiber laser ’

s radiation force ranged from 0.3 to3 μ N, below our verification range. Therefore, we have nodirect measurement of the scale ’s error in this range, butestimate it to be at least 2% based on the increased dis-agreement seen in the comparison over the 10 to 50 μ Nrange (Fig. 5). We expect that operating the scale in thenonstandard “ horizontal force ” orientation should have a negligible effect on its calibration due to supporting flex-ures operating independent of gravity, but so far haveonly observed this agreement to within a few percent(due to measurement noise). The uncertainty on the mea-sured mirror reflectivity is only a few tenths of a percent.We estimate the total measurement uncertainty to be onthe order of several percent, which is commensuratewith the measured 7% disagreement with theory. Morework is required to determine the measurement uncer-tainty down to the 1% level.

For the CO 2 laser, the true power was known only to a 6% uncertainty, and the incidence angle known only to5° (8% amplitude uncertainty) yielding a total uncertaintyof 10%, which roughly agrees with the 13% discrepancymeasured.

Although radiation force is commonly used as a tool inapplications ranging from remote force application [ 16]to laser cooling [ 17] to “ tractor beams ” [18], its accuratemeasurement is often obscured by “ radiometric ” forceswherein air flows from the cold side of the mirror to

the hot, creating a secondary force that does not followfrom Eq. ( 1) [19]. In the past, these radiometric effectshave been reduced by operating the measurement in a vacuum. However, this is not practical for our applicationas a robust power meter, and in fact, we find that our measurement conditions render such radiometric effectsnegligible. We demonstrate the dominance of true radia-tion forces below, but hypothesize several reasons for the lack of significant radiometric forces. First, our high-

reflectance mirrors allow only a very small fraction(0.002 to 0.003) of the incident light to be available for conversion to heat in the mirror itself. Furthermore, theglass-substrate dielectric mirror (1071 nm) transmits themajority of the unreflected light with little absorption. And in the 25 mm dielectric mirror and the 200 mm Sisubstrate mirror, the beam underfills the mirror, meaningany differential (front-to-back) heating that does occur will be relatively far from the mirror edge where thiseffect takes place, reducing the effect. Nor does the curi-ous rise in measured force for the CO 2 laser (beginning just before 5 s in Fig. 4) seem to be a radiometric, effectsince the ∼15 s duration of the effect should be more

than sufficient to establish equilibrium between the mir-ror front and back, eliminating the molecular flow.We demonstrate empirically the lack of radiometric

forces through two experiments. First, since photon mo-mentum is a vector quantity and the radiometric heatingeffect is scalar (independent of the light ’s angle of inci-dence), the radiation force imparted to a mirror can beidentified by its cosine dependence on the incident angle.We launched ∼525 W of optical power (1071 nm) ontothe dielectric mirror at incidence angles from ∼7 ° to 45°.The measured force (Fig. 6) agrees well with the ex- pected cosine behavior, indicating that the measuredforce is due to radiation pressure and not a heating effect.

As a second means of verifying radiation pressure asthe source of the measured force, we drastically reducethe incident optical power while increasing the absorbedoptical power. This will reduce the radiation pressure sig-nal but increase any thermal (radiometric) effects. Wereplace the mirror of Fig. 1 with an ∼80% absorbing disc(BK 7 glass flat coated with a carbon nanotube absorbinglayer). We direct 4.6 W of modulated laser light (1071 nm)onto the absorber. The 3.7 W of absorbed power is equiv-alent to the amount of light available for absorption if 1.2 kW were incident on our dielectric mirror. If themodulation seen in Fig. 2 is due to an absorption-basedmechanical effect, we should see more than twice the

80

60

40

20

0

P o w e r

( k W )

2520151050-5

Time (s)

Fig. 4. CO 2 laser power versus time (92 kW injected power)measured by radiation pressure. The solid curve is scale outputcorrected for incidence angle and mirror reflectivity; shadedregion indicates injection amplitude and duration.

-2.0-1.5-1.0-0.50.0

E r r o r

( % )

0.001 0.01 0.1 1Mass (g)

Fig. 5. Scale verification showing error as percent disagree-ment of scale readings with calibrated masses. Repeatedmeasurements indicate measurement variability.

3.6

3.2

2.8

2.4

2.0 M e a s u r e d

f o r c e

( N )

50403020100Angle of incidence (degrees)

Fig. 6. Measured force amplitude (solid circles) for variousangles of incidence demonstrating expected cosine depend-ence (solid line).

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effect here. In fact, we see no signal at the 0.1 Hz modu-lation frequency (Fig. 7). These results demonstrate thatthe modulation seen in Figs. 2 and 3 is indeed due toradiation force and not radiometric effects and sets anupper limit to the size of such effects at 17 dB belowthe radiation force signal (calculated from the signal-to-noise ratio of Fig. 7 plus 3.8 dB to scale the signal levelto the effective 1.2 kW).

Notably, the discrepancies in Fig. 3 have the same signfor both the CO 2 and the fiber laser and are on the sameorder as the measurement uncertainties. If this disagree-ment indicates a bias, it could be due to laser-based heat-ing of the scale itself or static build-up on the mirror.In the CO 2 case, the 10.6 μ m light experiences someatmospheric absorption and could be generating a heated air column around the beam, forcing the mirror slightly. Such possibilities will be examined more care-fully in the future as we seek to reduce the measurementuncertainty.

Using the signal and noise spectra of Fig. 7 andknowing the peak at 0.1 Hz corresponds to a 3.3 μ N force,we identify this scale ’s dynamic force noise floor as4 μ N⁄ Hz 1 ⁄ 2 at 0.1 Hz modulation frequency or, in termsof optical power, 600 W ⁄ Hz 1 ⁄ 2 . This picture of the noisefloor allows investigation of other applications. Thenominal 1 ⁄ f frequency dependence of the noise floor (above ∼0 .1 Hz) makes it attractive to consider a reversecalibration (calibrate scale response by using optical power). Generally milligram-level scale calibration is a difficult process involving a static “ weighing ” of a knownmass at zero frequency where the noise floor is a maxi-mum. This “ DC ” measurement requires that calibrationsbe done under strict environmentally stable conditions.However, if the reference “ mass ” were instead appliedas modulated laser power, the signal-to-noise ratio could

be instantly improved by a factor proportional to theratio of modulation frequency to measurement band-width. This could provide rapid and high-accuracymeasurements of scale linearity by use of the laser asa well-characterized force dither much smaller thanavailable from calibrated masses.

Targeting industrial use, high-power laser weldingapplications can require laser pulse-energies known towithin 1%. An optical energy meter capable of accuratelymeasuring individual laser pulses (e.g., 10 J⁄ pulse and10 Hz repetition rate) would be useful. Again, assuminga simple 1 ⁄ f noise extrapolation and appropriate filter-ing, the scale demonstrated here would need an approx-imately 13 dB reduction in noise level to perform such a measurement. However, commercial scale technology is

not optimized for AC measurements, and such improve-ments should come relatively easily.

We have shown feasibility of optical radiation-forcemeasurement of laser power using a commercial scale.Future investigations will carefully determine the ulti-mate accuracy of this technique and the modificationsneeded to enable other applications.

This publication of the U.S. government is not subjectto U.S. copyright. The authors thank Z. Kubarych, G.Shaw, and J. Pratt for helpful discussion and J. Bagfordand D. Siebert for use of the high-power laser facility.

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0.0001

0.001

0.01

0.1

1

A m p

l i t u d e

( a . u . )

0.012 3 4 5 6 7

0.12 3 4 5 6 7

1 Fourier frequency (Hz)

Fig. 7. Force spectrum from scale with 0.1 Hz modulated laser power, 533 W incident on mirror (black), and 4.6 W incident onabsorber (light red). With no signal, we treat the absorber curveas our noise floor.

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