---
title: Electromagnetic Momentum
module: Maxwell’s Equations and Electromagnetic Waves
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  A light beam carries no mass, yet it pushes: shine it on a surface and the surface feels a
  force. We trace that force back to the fields, which store energy with density
  $\varepsilon_0E^2$ and carry it along the Poynting vector
  $\vec S=\vec E\times\vec B/\mu_0$. Because that energy also carries momentum $U/c$, an
  absorbed beam presses with $I/c$ and a mirror with $2I/c$. We derive the Poynting theorem
  as local energy conservation, tie intensity to field amplitude, and work the momentum
  balance carefully enough that oblique incidence, partial reflection, and finite beams all
  drop out of one accounting.
topics: [Maxwell’s Equations and Electromagnetic Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 30 — Maxwell’s Equations and Electromagnetic Waves; §30-4"
---

## Field Energy and Poynting Flux

Electric and magnetic fields store energy locally. In vacuum, their instantaneous
energy densities are

$$
u_E=\frac{\varepsilon_0E^2}{2},
\qquad
u_B=\frac{B^2}{2\mu_0}.
$$

An ideal plane electromagnetic wave has $E=cB$. Substitution of
$c^{-2}=\mu_0\varepsilon_0$ gives

$$
u_E=u_B,
\qquad
u=u_E+u_B
=\varepsilon_0E^2
=\frac{B^2}{\mu_0}.
$$

The equality holds point by point for a vacuum plane wave. Near an antenna, capacitor,
waveguide wall, or material boundary, electric and magnetic energy densities can
differ strongly. The plane-wave relation must be established from geometry and field
data before it is used to combine energy terms.

$$
% caption: Energy split in a vacuum plane wave. Because $E=cB$, the electric density
% $\varepsilon_0E^2/2$ and the magnetic density $B^2/2\mu_0$ are equal at every point,
% so the total $u=\varepsilon_0E^2$ is exactly twice either half.
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The time average of a squared sinusoid is one half its peak square. With
$E(t)=E_0\cos(kx-\omega t)$, the average energy density is

$$
\langle u\rangle
=\frac{\varepsilon_0E_0^2}{2}
=\varepsilon_0E_{\rm rms}^2.
$$

The RMS convention should be identified whenever energy, intensity, or pressure is
computed from a measured field. A field meter can report peak, RMS, or detector-scaled
amplitude. Mixing a peak electric amplitude with an RMS magnetic amplitude changes the
result by a factor of $\sqrt2$ or two, depending on the formula.

Energy density alone does not specify transport direction. A standing wave can have
large local field energy while its time-averaged energy flux vanishes. A source-near
field can store and return energy over each cycle without carrying the same energy to
infinity. The flux calculation therefore requires both electric and magnetic
components with their directions and relative phase.

### The Poynting theorem

Take the dot product of the Ampère–Maxwell equation with $\vec E$ and the dot
product of Faraday's law with $\vec B/\mu_0$. Combining the two with the vector
identity for the divergence of a cross product gives the local energy balance

$$
\frac{\partial}{\partial t}
\left(
\frac{\varepsilon_0E^2}{2}
+\frac{B^2}{2\mu_0}
\right)
+\nabla\mathbin{\cdot}\vec S
=-\vec J\mathbin{\cdot}\vec E,
$$

where

$$

\vec S=\frac{1}{\mu_0}\vec E\mathbin{\times}\vec B

$$

is the Poynting vector. The term $-\vec J\mathbin{\cdot}\vec E$ transfers
energy between fields and charges. In a source-free vacuum region it is zero, so local
field-energy change is balanced by the divergence of energy flux.

In a plane wave, $\vec E$, $\vec B$, and $\vec S$ form a
right-handed triad. The Poynting-vector magnitude is

$$
S=\frac{EB}{\mu_0}
=\varepsilon_0cE^2.
$$

The instantaneous flux oscillates at twice the field frequency because it contains a
product of in-phase sinusoidal fields. Its cycle average is the intensity:

$$
I=\langle S\rangle
=\frac{E_{\rm rms}B_{\rm rms}}{\mu_0}
=\frac{E_0B_0}{2\mu_0}
=c\langle u\rangle.
$$

Intensity has units $\mathrm{W\,m^{-2}}$. Its definition requires a surface normal
to the local propagation direction. A slanted detector of area $A$ intercepts
power $IA\cos\theta$ in the uniform-beam approximation, where $\theta$ is the
angle between detector normal and $\vec S$.

$$
% caption: Field triad of a plane wave. The transverse electric and magnetic vectors
% and the Poynting vector $\vec S=\vec E\times\vec B/\mu_0$ form a right-handed set, so
% $\vec S$ points along the propagation direction, perpendicular to both fields.
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The integral form over a fixed volume $V$ is

$$
\frac{\d}{\d t}\int_Vu\,\d V
=-\oint_{\partial V}\vec S\mathbin{\cdot}\d\vec A
-\int_V\vec J\mathbin{\cdot}\vec E\,\d V.
$$

It distinguishes energy that enters a volume through its surface from energy converted
to mechanical motion, internal excitation, or heat within the volume. A resistor
absorbs electromagnetic energy through fields around its conductors; a capacitor
stores field energy in its gap; a source delivers energy to fields through charge work.
Circuit symbols omit those spatial paths, but the integral theorem keeps the energy
accounting consistent across the entire apparatus.

## Intensity and Energy Measurement

An intensity measurement needs an active area, an orientation, a response spectrum,
and a calibration traceable to power or field. A thermal detector estimates absorbed
power from temperature change or electrical substitution. A photodiode produces a
current related to photon absorption and responsivity. An antenna or pickup loop
measures a field-linked voltage that requires an effective-area or effective-height
calibration before conversion to intensity.

Power is the surface integral

$$
P=\int_A\vec S\mathbin{\cdot}\d\vec A.
$$

A broad uniform beam normal to a detector has $P=IA$. A Gaussian beam, a focused
beam, a spherical wave, or a partially blocked aperture needs the measured spatial
distribution. Scan the detector across the beam with a stated step size, correct
background and detector nonlinearity, then integrate the corrected normal flux over
area. A central maximum multiplied by aperture area overestimates a beam whose edges
carry lower intensity.

### Flux maps and pulse energy

The Poynting vector is a local quantity. A two-dimensional map of $\vec E$ and
$\vec B$ can be converted into a map of $\vec S$ only after both component
directions and relative phases are known. A scalar intensity detector gives one
projection of time-averaged flux. It cannot determine whether local energy circulates,
returns from a boundary, or crosses the detector plane at an oblique angle. Vector
field probes or a model constrained by component measurements are needed when the
direction of energy transport matters.

A pulse crossing a plane of area $A$ has transmitted energy

$$
U_{\rm pulse}
=\int_{-\infty}^{\infty}
\left[
\int_A\vec S(\vec r,t)
\mathbin{\cdot}\d\vec A
\right]\d t.
$$

A plane pulse with uniform transverse area and duration $\Delta t$ has
$U_{\rm pulse}\simeq IA\Delta t$ when $I$ is the time-averaged intensity over a
carrier cycle. A few-cycle pulse needs its instantaneous Poynting flux integrated
directly; the carrier-average approximation can obscure envelope edges and detector
bandwidth effects.

Field-energy maps need a volume convention. A numerical grid often stores electric
and magnetic components at staggered positions and times. Interpolating both to a
common cell center before forming $E^2$, $B^2$, and $\vec E\times\vec B$
avoids artificial checkerboard patterns. The cell energy and net flux through its
faces should satisfy a discrete Poynting balance up to source, loss, and numerical
truncation terms. A displayed arrow field can look smooth while its cell-by-cell
energy balance fails.

The volume integral exposes sign errors. If more Poynting flux enters a source-free
cell than leaves during a time interval, stored field energy must increase. If net
outgoing flux exceeds incoming flux, stored energy must decrease. A negative
$\vec J\mathbin{\cdot}\vec E$ term means charges deliver energy to the
fields; a positive term means fields deliver energy to charges. The sign depends on
the definition of conventional current and electric field, so a circuit current arrow
must be reconciled with the local $\vec J$ direction before power is interpreted.

Time averaging requires a stated window. A continuous sinusoid has a period average
that removes the $2\omega$ oscillation in $S$. A modulated signal has an envelope
whose average depends on window length. A pulse train needs averaging over an integer
number of repetitions or an explicit duty-cycle factor. A detector with a thermal
time constant reports a long average; a fast electro-optic sensor can resolve carrier
or envelope structure. Matching theoretical average to detector response prevents a
comparison of unlike quantities.

A pulse-energy calibration may use electrical substitution. Deliver known electrical
energy to a thermal absorber, establish the detector response, then expose the same
absorber to the electromagnetic pulse under identical thermal conditions. Detector
linearity must be checked across the pulse-energy range. Window transmission, surface
reflection, aperture clipping, and absorber emissivity change the amount of incident
energy that becomes the calibrated detector signal. Those factors belong in a
throughput model, not in an unexplained detector constant.

Beam maps require sampling density tied to the narrowest spatial structure. A wide
Gaussian beam may be represented by a coarse grid near its edges and a fine grid near
its center. A beam with interference fringes, speckle, or aperture diffraction needs
sampling fine enough to resolve the fringe spacing. Repeat the area integral after
refining the scan. The change in integrated power provides a direct resolution
uncertainty; a visually smooth color map provides no comparable error estimate.

The field-energy description also separates propagation loss from detector loss. A
drop in measured power between two planes can arise from beam divergence, material
absorption, scattering, reflection, or a detector aperture that clips the beam.
Surveying the transverse flux at both planes identifies whether total flux has fallen
or merely spread beyond one detector. The corresponding model must include the same
surface area at both planes.

## Momentum and Radiation Pressure

Electromagnetic energy carries momentum. In vacuum, the momentum density is

$$
\vec g=\frac{\vec S}{c^2},
$$

and a pulse of total energy $U$ traveling in one direction carries momentum magnitude

$$
p=\frac{U}{c}.
$$

The result follows from the field momentum density and applies to the net energy that
crosses a surface. A travelling wave transfers momentum to matter when it is absorbed,
reflected, scattered, or redirected. The transferred momentum appears as mechanical
force, deformation, or a support reaction. The local force distribution depends on
beam shape, material response, and surface geometry.

For normal incidence on a perfectly absorbing surface, momentum flux per unit area is

$$
p_{\rm rad}=\frac{I}{c}.
$$

The symbol $p_{\rm rad}$ denotes radiation pressure, with units
$\mathrm{N\,m^{-2}}$. A perfectly reflecting mirror reverses the normal momentum
of the wave, giving

$$
p_{\rm refl}=\frac{2I}{c}.
$$

An opaque surface with absorptance $A$ and reflectance $R$, where $A+R=1$, has
normal-incidence pressure $(A+2R)I/c$ in the ideal specular model. Diffuse
reflection, transmission, surface roughness, and heating-driven motion require a
momentum balance using the actual outgoing angular distribution.

$$
% caption: Momentum transfer at normal incidence. An absorber stops the beam and feels
% pressure $I/c$; a mirror also sends the beam back, reversing its momentum, so the
% momentum change and the force double to $2I/c$ (drawn as a longer force arrow).
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$$

Oblique incidence introduces an area projection and a momentum projection. Let
$\theta$ be the angle between the incoming wave direction and surface normal. The
incident power crossing one unit area of the physical surface is $I\cos\theta$. Its
normal momentum fraction adds another $\cos\theta$. An absorbing plane therefore
has normal pressure

$$
p_{\perp,\rm abs}
=\frac{I\cos^2\theta}{c}.
$$

An ideal specular reflector doubles that normal pressure. Tangential momentum remains
with the reflected beam in the smooth-specular model; a rough or absorbing surface can
receive tangential momentum and experience a lateral force. The surface normal, beam
direction, and outgoing directions belong in the force diagram.

$$
% caption: Oblique incidence. The beam meets the surface at angle $\theta$ to the
% normal. Its projected power scales as $\cos\theta$ and its normal momentum fraction
% as another $\cos\theta$, so an absorber feels normal pressure $I\cos^2\theta/c$; a
% specular reflection reverses the normal momentum and doubles it.
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Radiation pressure is small for ordinary room illumination. An intensity of
$1000\ \mathrm{W\,m^{-2}}$ gives an absorbing-surface pressure of about
$3.3\ \mathrm{\mu Pa}$. The force becomes measurable with large area, high
intensity, low mechanical stiffness, or long averaging time. A laser beam can exert a
detectable force on a lightweight mirror, while thermal expansion, air currents, and
electrostatic attraction can easily exceed the radiation force in an unshielded setup.

$$
% caption: Pressure against intensity for the two ideal targets at normal incidence.
% Both are straight lines through the origin; the reflecting slope $2/c$ is exactly
% twice the absorbing slope $1/c$.
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$$

Force measurement uses an independent mechanical calibration. A torsion balance,
flexure, optical lever, or force transducer converts displacement into force through a
stated stiffness or transfer function. Modulate beam intensity at a frequency away
from ambient vibration, then demodulate the mechanical response with the same
reference. A beam shutter test measures background drift. Reversing a mirror
orientation or exchanging absorber and reflector targets tests the predicted momentum
change while retaining the mechanical apparatus.

Beam momentum can also be inferred from force balance on an absorbing volume. A
calorimeter measures absorbed power, while a force sensor measures the momentum rate.
The ratio $F/P$ should approach $1/c$ for a stationary opaque absorber in vacuum
after accounting for reflected and transmitted power. Measuring both quantities with
separate calibrated instruments is stronger than inferring pressure from source power
alone, because source-to-target coupling and aperture loss can be measured directly.

## Near Fields and Scattering

Field-energy flow close to a source differs from far-zone radiation. A driven antenna
or oscillating dipole stores electric and magnetic energy in its surrounding region.
During one part of a cycle, source work increases that stored energy. During another
part, some stored energy returns to the source circuit. The instantaneous Poynting
vector can point inward, outward, or circulate locally. A nonzero instantaneous flux
therefore does not by itself establish net radiation to infinity.

Integrate Poynting theorem over a closed surface surrounding a source. The outward
power is

$$
P_{\rm out}(t)
=\oint_{\partial V}\vec S\mathbin{\cdot}\d\vec A.
$$

Average a periodic source over a full cycle. The mean outward flux through a surface
in the far zone is radiated power. A small surface close to the feed can have
large oscillatory flux associated with field storage. Its cycle average may include
source loss, conductor loss, dielectric loss, and power delivered to nearby objects.
The measurement surface and its distance from the source should be part of every
reported radiated-power value.

$$
% caption: Energy flow around a driven source. On the near boundary the instantaneous
% Poynting flux reverses within a cycle as stored energy returns to the source, while
% across the far boundary the cycle average is a net outward radiated power.
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The complex Poynting vector offers a frequency-domain description of this distinction.
For sinusoidal phasors under one stated convention,

$$
\widetilde{\vec S}
=\frac12\widetilde{\vec E}
\mathbin{\times}\widetilde{\vec H}^{\,*}.
$$

Its real part gives cycle-averaged power flow. Its imaginary part describes
oscillatory energy exchange associated with reactive fields. The factor one half
comes from phasor averaging. Using peak phasors in one equation and RMS values in
another changes the numerical factor. The $\vec H$ field belongs in the
macroscopic material form; in vacuum $\vec H=\vec B/\mu_0$.

The reactive component is large near many sources and resonators. It can change sign
with position or frequency without violating energy conservation. A detector that
measures only absorbed average power responds mainly to the real power flow. A
field-sensitive probe can respond to stored energy and require careful interpretation.
Separating real and imaginary Poynting components prevents a local high field from
being labeled as a high radiated intensity.

Source efficiency compares radiated power with input power. A source can draw
substantial current while radiating little if it has large conductor loss, dielectric
loss, mismatched feed power, or energy returned to the drive circuit. Measure input
power at a stated reference plane, measure or calculate loss paths, and integrate
far-zone intensity over a closed surface or a sufficiently sampled angular pattern.
The difference between input and radiated power should be assigned to measured loss or
uncertainty, not left as an unexplained efficiency gap.

Angular intensity maps connect local Poynting flux with total radiated power. For an
axisymmetric far-field pattern with intensity $I(\theta)$, integrate

$$
P_{\rm rad}
=2\pi r^2\int_0^\pi I(\theta)\sin\theta\,\d\theta.
$$

The factor $r^2$ converts intensity at radius $r$ to power per solid angle.
Uniform sampling in polar angle does not correspond to uniform solid-angle sampling
because of the $\sin\theta$ weight. A detector scan should therefore retain angular
coordinates and solid-angle factors before a total power is estimated.

$$
% caption: Far-zone angular integration. A detector scanned over polar angle $\theta$
% at fixed radius samples the intensity pattern; each ring carries a solid-angle weight
% $\sin\theta$, so $P_{\rm rad}=2\pi r^2\int_0^\pi I(\theta)\sin\theta\,\d\theta$ and a
% peak direction alone does not fix the total.
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The source boundary also determines what counts as input work. A battery, RF
amplifier, optical pump, or charged capacitor can supply energy through different
mechanisms. The Poynting theorem handles each case when the volume includes the
relevant charges and materials. A source circuit diagram alone does not determine how
much power becomes radiated flux; field and loss measurements establish that connection.

Near-field probes perturb the field they sample. A metal loop changes local magnetic
boundary conditions; an electric dipole probe loads the electric field; a calorimeter
absorbs energy and can shadow a beam. Repeat a source measurement with probe distance,
orientation, and size varied. A stable extrapolation to a small or distant probe
supports a nonperturbative interpretation. A probe-induced change in source current or
far-zone pattern belongs in the measurement uncertainty.

### Scattering and momentum balance

Radiation pressure is a surface-momentum balance. An incident beam carries a momentum
flux vector $\vec S/c^2$. Each reflected, transmitted, absorbed, or scattered
portion carries an outgoing momentum flux. The net force on an enclosed target equals
the incoming momentum rate minus the outgoing momentum rate, together with any change
in electromagnetic momentum stored inside the enclosing surface. The balance remains
valid for a curved target, a refracting particle, or a rough surface; simple
$I/c$ formulas represent special geometries.

The Maxwell stress tensor gives a field-based surface calculation in vacuum:

$$
\vec T
=\varepsilon_0
\left(
\vec E\vec E-\frac12E^2\vec I
\right)
+\frac{1}{\mu_0}
\left(
\vec B\vec B-\frac12B^2\vec I
\right).
$$

A closed surface surrounding matter gives force from stress flux and the time rate of
field momentum inside the volume:

$$
\vec F_{\rm matter}
=\oint_{\partial V}\vec T
\mathbin{\cdot}\d\vec A
-\frac{\d}{\d t}\int_V\vec g\,\d V.
$$

The tensor expression becomes valuable when incident and outgoing beams have several
directions or when a target bends a beam. It avoids assigning a scalar pressure to a
surface whose local normal and local field direction vary across the illuminated area.

$$
% caption: Momentum-balance surface around a scatterer. All incoming and outgoing
% Poynting flux crosses one closed vacuum boundary; the target force equals the net
% momentum flow through it plus any change in field momentum stored inside, so no
% scalar pressure need be assigned to the surface.
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Absorption changes both mechanical force and thermal state. A blackened target can
absorb most incident optical power, increasing radiation momentum transfer and
heating. Heating can expand a mount, generate convection, or change material
reflectance. A force measurement that follows beam modulation at a thermal time
constant can contain photothermal motion as well as radiation pressure. Modulate at
several frequencies: direct momentum force follows the mechanical transfer function,
while thermal force often carries an additional slow phase lag and amplitude roll-off.

Specular reflection preserves beam coherence and yields a well-defined outgoing
direction. Diffuse reflection distributes outgoing momentum over many angles. A
Lambertian surface may have the same reflected power as a mirror while delivering a
different normal force. Measure angular power distribution when target roughness,
surface texture, or scattering medium prevents a specular model. Integrating only
total reflected power loses the angular momentum information required for force.

$$
% caption: Specular versus diffuse reflection. A mirror sends the beam into one
% direction; a rough surface spreads comparable reflected power over many angles. The
% normal force follows the vector sum of the outgoing momenta, so equal reflected power
% can give unequal force.
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Transmission through a transparent object also transfers momentum. Refraction changes
the direction of the transmitted beam, so the object receives the opposite transverse
momentum change. A symmetric beam can produce zero net lateral force while creating
internal stress. Optical trapping and beam steering use such momentum redirection.
The surface-force calculation must use the incident and transmitted media consistently;
the field momentum convention in material media requires careful treatment of
polarization and magnetization. Vacuum external-surface measurements provide a clear
way to determine the total force without selecting an internal momentum partition.

Beam shape affects torque as well as net force. A centered symmetric beam on a
symmetric target has zero torque about its center. A beam displaced by lever arm
$\ell$ produces torque approximately $F\ell$ when the force direction is normal
to the target. A spatial intensity map and target coordinate survey identify the
center of pressure. Torsion-balance measurements must distinguish a torque generated
by beam offset from an apparent force generated by a linear displacement sensor.

The target may move appreciably during a long measurement. A moving mirror Doppler
shifts reflected light and changes the mechanical power balance. For a slowly moving
target, radiation force is still set primarily by the incident momentum flux, but
mechanical work $Fv$ draws energy from the beam. A full moving-boundary calculation
is needed when target speed is large enough to change frequency, angle, or source
coupling over the measurement interval.

Mechanical resonance can magnify a small radiation force. A compliant target driven
near its natural frequency has a displacement set by force, damping, stiffness, and
drive phase. Calibration with an electrostatic or magnetic actuator at the same
frequency tests the mechanical transfer function. The calibration force must act at
the same location and direction as the beam force or a mode-shape difference can
create a scale error.

## Beam and Force Measurements

Many laboratory beams have an approximately Gaussian transverse profile,

$$
I(r)=I_0\exp\left(-\frac{2r^2}{w^2}\right),
$$

where $w$ is the radius at which intensity has fallen to $e^{-2}$ of its axial
value. Integrating over the full transverse plane gives

$$
P=\frac{\pi w^2I_0}{2}.
$$

The relation connects a local intensity measurement to total beam power only when the
profile is known and the beam is adequately captured. A detector centered on the axis
with radius $R$ intercepts

$$
P(R)=P\left[1-\exp\left(-\frac{2R^2}{w^2}\right)\right].
$$

A detector aperture smaller than the beam therefore measures a specified power
fraction, not the total emitted power. A clipped beam can still supply a reliable
local intensity if its active area and position are documented.

$$
% caption: Gaussian beam and aperture capture. Left: intensity falls as
% $\exp(-2r^2/w^2)$ from the axis. Right: a detector of radius $R$ collects the
% fraction $1-\exp(-2R^2/w^2)$, approaching total power only as $R$ grows past the
% beam width.
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$$

> **Worked example (Gaussian beam intensity and force).** A Gaussian beam carries total
> power $P=5.00\ \mathrm W$ with width $w=4.00\ \mathrm{mm}$. Its axial intensity is
>
> $$
> I_0=\frac{2P}{\pi w^2}
> =1.99\times10^5\ \mathrm{W\,m^{-2}}.
> $$
>
> A fully illuminated absorbing target receives force $P/c=16.7\ \mathrm{nN}$; a perfect
> reflector receives twice that. The axial pressure far exceeds the average over a wide
> target because the Gaussian energy is concentrated at small radius, so multiplying $I_0$
> by a target area is valid only for a target smaller than the region where the intensity
> stays nearly uniform.

Power calibration and profile calibration are separate measurements. A calibrated
thermal power meter can set the total beam scale. A camera or scanning detector can
set relative spatial intensity after correcting pixel response, saturation, dark
signal, and optical attenuation. Normalize the spatial map so that its area integral
equals the independently measured total power. This combined approach exposes a
camera gain drift or a power-meter aperture loss that neither measurement alone
identifies.

Detector responsivity may depend on wavelength, polarization, incidence angle, and
temperature. A power meter calibrated at one wavelength cannot be assumed to have the
same response at another wavelength. An optical window can add reflection that varies
with angle and polarization. A detector placed in a converging beam can sample a
different angular distribution from the calibration beam. Record wavelength, beam
diameter, polarization, aperture, and calibration standard in the radiometry log.

Pulsed beams need energy and repetition rate. A detector that reports average power
$\bar P$ at repetition rate $f_{\rm rep}$ gives pulse energy

$$
U_{\rm pulse}=\frac{\bar P}{f_{\rm rep}}
$$

only when pulses are identical and background has been removed. Peak intensity further
depends on pulse duration and temporal shape. A rectangular-pulse approximation uses
$I_{\rm peak}\simeq U_{\rm pulse}/(A\Delta t)$; Gaussian temporal and spatial
profiles require both integrals. Reporting average power alone does not determine
peak radiation pressure or peak field amplitude.

An energy-balance experiment can use three readings: source power, transmitted power,
and absorbed power. For a target with negligible scattering outside the measured
channels,

$$
P_{\rm in}=P_{\rm refl}+P_{\rm trans}+P_{\rm abs}.
$$

Measure each channel with a detector that has a compatible calibration and aperture.
The residual gives a closure test. A residual that changes with target angle can
indicate uncollected scatter; a residual that grows with source power can indicate
detector nonlinearity or heating. An energy-balance result supports a momentum
calculation because the same incident, reflected, and transmitted channels determine
the outgoing momentum flux.

Uncertainty propagation should retain correlation. The same power-meter calibration
can scale incident and transmitted readings together. A shared aperture-radius error
can change every point in a beam map. Detector noise can be independent from point to
point, while source-power drift correlates an entire scan. Present total power,
captured fraction, intensity scale, and radiation force with the calibration model
used to combine those effects.

> **Worked example (force on an opaque target).** A beam delivers measured incident power
> $P_{\rm inc}$ to an opaque target with absorptance $A$ and specular reflectance $R$
> (with $A+R=1$ and negligible transmission). The predicted normal force is
>
> $$
> F_{\rm pred}
> =\frac{(A+2R)P_{\rm inc}}{c}.
> $$
>
> For $P_{\rm inc}=5.00\ \mathrm W$, $R=0.90$, and $A=0.10$,
>
> $$
> F_{\rm pred}
> =\frac{(0.10+2(0.90))(5.00\ \mathrm W)}
> {2.998\times10^8\ \mathrm{m\,s^{-1}}}
> =31.7\ \mathrm{nN}.
> $$
>
> The power that counts is the power delivered to the target, not the nominal laser output:
> window loss, aperture loss, or beam spill all change $P_{\rm inc}$, and a rough target
> spreads the reflected momentum over many angles, altering the factor $A+2R$.

The force prediction should therefore be attached to measured channel powers and an
explicit surface-scattering model.

Measured force may be represented as a complex response under beam modulation.
Suppose beam power is modulated at angular frequency $\omega_m$, and a calibrated
mechanical transfer function $H_{\rm mech}(\omega_m)$ relates force to sensor
output. Dividing the sensor phasor by $H_{\rm mech}$ produces a force estimate with
amplitude and phase. The predicted radiation force is in phase with absorbed or
reflected optical power at the target, apart from the mechanical response. A delayed
thermal expansion signal has an additional material time constant and can be separated
by a frequency sweep.

Mechanical calibration should span the beam-force range. A large calibration force
applied to a flexure can shift its stiffness or excite a different mode. A calibration
at one frequency may not transfer to a beam modulation near resonance. Compare the
beam force estimate against at least two actuator amplitudes and several modulation
frequencies. Store displacement records and phase references; an amplitude-only
comparison cannot distinguish momentum force from delayed heating.

Residual patterns diagnose incomplete energy or momentum accounting. A force residual
proportional to incident power but with the wrong slope can indicate an incorrect
reflectance, intercepted-power fraction, or mechanical gain. A residual that depends
on target angle can indicate omitted specular or transmitted momentum. A residual
that follows beam position can indicate torque coupling or aperture clipping. A
residual that remains with the beam shuttered belongs to mechanical drift, electrical
pickup, or background light.

Plot residuals against incident power, target angle, modulation frequency, beam
position, and time. Fit one parameter only when a physical calibration independently
constrains the others. A flexible fit can force a force-versus-power line through the
data while hiding a target reflectance error inside a gain factor. The data record
should retain channel powers, detector calibration, mechanical displacement, source
reference, temperature, and target orientation for every run.

The measurement should include a shuttered null configuration. Terminate the source
or block the beam upstream of the target and record detector and mechanical channels
under the same timing and modulation procedure. Background can contain electrical
pickup, actuator leakage, ambient light, vibration, or thermal drift. Subtracting a
single mean value without retaining the background spectrum can create an apparent
force at the modulation frequency. Report the null amplitude and phase beside the
beam-on result.

## Limits and Time-Resolved Transfer

The vacuum Poynting vector gives a direct energy-flow description when fields are
measured in free space around a target. Matter adds polarization, magnetization, and
mechanical stress. A dielectric-filled region can store material energy in addition
to $\varepsilon_0E^2/2$. A magnetic material can exchange energy with its internal
magnetization. A dispersive material can store energy in a frequency-dependent
response. The total force on a body can still be measured from external momentum flow,
while partitioning momentum between fields and material inside the body requires a
specified macroscopic convention.

A force balance enclosing the entire target avoids many internal-model ambiguities.
Place the integration surface in surrounding vacuum, include all incident and outgoing
beams, and account for any electromagnetic momentum that changes inside the volume.
The external surface then supports a measurement of total force regardless of whether
the target is absorbing, refracting, magnetic, or structured. The surface must be far
enough from evanescent source-near fields and close enough that unrelated objects do
not cross it.

Finite beams require a local propagation direction. A focused beam has a spread of
wave vectors. Its Poynting vector can have transverse components, especially near a
focus or aperture. The scalar normal-incidence pressure formula gives a reference
estimate at a small locally plane patch. A full beam-force calculation integrates
vector momentum flux over the illuminated surface and uses the angular distribution of
outgoing radiation. A target much larger than the beam can simplify the capture
fraction; a target comparable to beam width needs a spatial overlap integral.

Surface temperature and environmental medium set practical limits. In air, absorption
heats the target and surrounding gas. Convection can exert a force much larger than
radiation pressure at low modulation frequency. Acoustic pressure from a pulsed laser
can move a target through a separate mechanical pathway. Measurements in vacuum,
rapid modulation, beam-size variation, and target-material exchange help separate
these effects. Each control changes a physical mechanism and should be incorporated
into the uncertainty model rather than treated as a cosmetic repeat.

The source may fluctuate in pointing, polarization, spectrum, and power. A detector
sampling only one point of a moving beam can record apparent intensity fluctuations
without a change in total power. Monitor a fraction of the beam with a reference
detector, record beam position at the target plane, and normalize the force data only
after verifying that the reference detector response is linear and stable. A reference
monitor located before a lossy optical path does not detect downstream aperture
clipping or target-surface changes.

The mechanical target can also alter the optical geometry. A tilted mirror changes the
return-beam path. A translating particle moves through an intensity gradient. A
deforming membrane changes its local surface normal. In a strong feedback regime, beam
force, target position, and optical intensity must be solved together. A static
pressure calculation serves only as the small-displacement approximation about one
defined operating point.

Data reduction should preserve sign conventions. The Poynting vector direction,
target normal, mechanical positive displacement, sensor polarity, and modulation
reference all determine the reported force sign. A reflected beam reverses an outgoing
momentum component; a detector cable inversion reverses only an electronics sign.
Test the complete chain with a known mechanical displacement and a known optical
power change before attributing a phase inversion to radiation momentum.

Several independent checks support a final energy-and-momentum result:

- The integrated detector map agrees with a calibrated total-power reading within the
  stated aperture and calibration uncertainty.
- Incident, reflected, transmitted, absorbed, and scattered power channels close the
  energy budget within uncertainty.
- The mechanical force changes linearly with delivered target power in the low-power,
  small-displacement regime.
- Replacing an absorbing target with a high-reflectance target changes the predicted
  momentum coefficient in the measured direction.
- Shuttered, rotated, and displaced-beam controls bound background, torque, and
  thermal mechanisms.

The reported conclusion names the incident power at the target, beam profile, target
area and optical properties, field or detector calibration, averaging interval,
force-transfer calibration, surface-momentum model, and uncertainty components. It
also states whether the result concerns local intensity, integrated beam power,
cycle-averaged Poynting flux, pulse energy, absorbing pressure, or reflecting
pressure. These quantities share a common electromagnetic energy framework but answer
different experimental questions.

### Pulse impulse

Radiation pressure integrated over time gives impulse. A pulse of absorbed energy
$U_{\rm abs}$ delivered normally to a target transfers momentum

$$
J_{\rm abs}=\frac{U_{\rm abs}}{c}.
$$

An ideal reflected pulse transfers

$$
J_{\rm refl}=\frac{2U_{\rm refl}}{c}.
$$

The impulse result applies even when pulse duration is shorter than the target's
mechanical response time. The target can receive a brief momentum transfer and then
move slowly under inertia, suspension stiffness, and damping. A force sensor with
bandwidth below the optical pulse bandwidth may record the mechanical response rather
than the instantaneous pressure waveform.

A pulse train has average force equal to pulse impulse times repetition rate:

$$
\langle F\rangle=J_{\rm pulse}f_{\rm rep}.
$$

This relation agrees with average power divided by $c$ when pulses have equal energy
and the target absorbs them fully. A changing pulse energy, beam overlap, or
reflectance changes the force term on each pulse. Record the pulse-energy distribution
instead of assuming a nominal repetition-rate product represents the delivered
momentum.

$$
% caption: Pulse impulse and delayed response. A short pulse deposits force over a
% brief interval (left); a suspended target responds slowly afterward (right). The
% time integral of the force is the impulse $J$, compared with absorbed or reflected
% pulse energy divided by $c$.
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Impulse measurements offer an independent mechanical route to optical energy. A freely
suspended target with known mass $m$ acquires a velocity change
$\Delta v=J/m$ when suspension forces are negligible during the pulse. A torsion
pendulum can convert the impulse into an angular deflection. Both methods require a
background measurement because acoustic shock, electrical pickup, and source recoil
can produce a synchronized mechanical signal. A target swap between absorber and
mirror tests the expected momentum coefficient under the same pulse energy.

The pulse field can have a broad spectrum. Detector response and target reflectance
may vary across that spectrum, so a single wavelength calibration can misestimate
absorbed energy. Measure or bound spectral content, use a broadband calibrated energy
detector, and include spectral weighting in the target-power estimate. The force
formula remains an energy-momentum relation; the experimental task is determining the
energy and outgoing momentum channels that actually reached the target.

Consistency checks connect the energy and force records. The time integral of detected
incident flux should reproduce the independently measured pulse energy after aperture
and calibration corrections. The target-force impulse should scale with that delivered
energy and with the selected absorption or reflection coefficient. Reversing beam
direction reverses the mechanical impulse relative to the laboratory axis. Reducing
the target aperture lowers captured energy and force together, provided beam position
and target properties remain fixed. A mismatch in only one of these tests points to a
specific part of the model: detector integration, mechanical calibration, momentum
coefficient, or target overlap.

The same audit applies to continuous beams after replacing pulse energy with power and
impulse with time-averaged force. Energy, momentum, and pressure then form one
quantitative chain: measured fields or detector signals determine flux; flux
integrates to power; power and outgoing direction determine momentum rate; momentum
rate determines force on the target. Retaining each link in the data record prevents a
force result from being separated from the electromagnetic energy measurement that
supports it.

Unit checks provide a compact final screen. Energy density has units
$\mathrm{J\,m^{-3}}$, Poynting flux has units $\mathrm{W\,m^{-2}}$, momentum
density has units $\mathrm{N\,s\,m^{-3}}$, and radiation pressure has units
$\mathrm{N\,m^{-2}}$. Multiplying a flux by area and time gives energy; dividing
that energy by $c$ gives momentum. These dimensional links expose an omitted area,
an RMS-versus-peak factor, a missing time integral, or a pressure calculation that
used source power instead of delivered target power. They also keep local field
quantities distinct from detector-integrated quantities throughout a beam or pulse
analysis.

Reference-plane discipline completes the audit. State where source power is measured,
where beam power is measured, and where force is measured. Cable connectors,
windows, apertures, and optical elements between those planes can store, dissipate,
redirect, or clip energy. Each calibration factor therefore belongs to a named
transmission or response term between two reference planes. The resulting
energy–momentum balance can then be reproduced from the recorded measurements.