Converting Between Optical Energy Density, Intensity, Photon Flux, and Photon Rate

Hello, and welcome to the first module of this laser-physics refresher. We will begin by establishing a quantitative “unit-conversion language” for light: how electromagnetic-field energy transport, beam power and area, photon number, and spectral bandwidth describe the same radiation from different viewpoints.

This is foundational for the rest of the course. In the next lessons, the radiation field will drive absorption and stimulated-emission rates; there, choosing correctly between an intensity, a photon flux, and a spectral energy density will matter.

A typical study path for this lesson is about 40 minutes: a short electromagnetic-wave review, beam-geometry bookkeeping, then spectral and photon-number conversions.

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From electromagnetic fields to optical intensity

For a propagating electromagnetic wave in vacuum, electric and magnetic fields each carry energy. The instantaneous energy density is

$$u=\frac{1}{2}{ϵ}_0{E}^2+\frac{1}{2{\mu }_0}{B}^2.$$

For a plane traveling wave, $B=E/c$, so the electric and magnetic contributions are equal. The total instantaneous energy density is therefore

$$u={ϵ}_0{E}^2=\frac{B^2}{{\mu }_0}.$$

Energy is transported in the propagation direction. The relevant flux is the Poynting vector,

$$\mathbf{S}=\frac{1}{{\mu }_0}\mathbf{E}\times \mathbf{B},$$

whose magnitude has units of $\mathrm{W}{\mathrm{m}}^{-2}$. Optical detectors cannot follow field oscillations at hundreds of terahertz, so laser physics normally uses the cycle-averaged flux:

$$I\equiv ⟨S⟩.$$

This is optical intensity: power transmitted per unit area through a plane perpendicular to the beam.

For a monochromatic plane wave in vacuum with electric-field amplitude $E_0$,

$$I=\frac{1}{2}c{ϵ}_0{E}_0^2.$$

Thus, doubling field amplitude makes intensity four times larger. Frequency does not appear explicitly in this classical relation: at a fixed field amplitude, a red and a blue plane wave transport the same average energy flux. They differ in how that energy is partitioned into photons.

For a narrowband wave in a transparent, nonmagnetic dielectric of refractive index $n$, a commonly useful relation is

$$I=\frac{1}{2}c{ϵ}_{0}n{E}_0^2.$$

More generally, the energy-density relation for a directed beam is

$$I=v_{g}u,$$

where $v_g$ is the group velocity and $u$ is the cycle-averaged energy density. In vacuum, $v_g=c$, giving $I=cu$. In most introductory laser calculations involving a weakly dispersive transparent material, $v_g$ is approximated by $c/n$. Keep the more general form in mind if strong dispersion becomes relevant.

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Beam geometry: local intensity versus total power

Intensity is a local quantity. Power is the total energy-transfer rate across the beam:

$$P={\int }_{A}I(\mathbf{r})dA.$$

The cross-sectional plane must be perpendicular to propagation. A useful dimensional check is

$$\left[\frac{P}{A}\right]=\mathrm{W}{\mathrm{m}}^{-2}.$$

Two transverse beam profiles occur constantly in laser work.

Flat-top beam

A flat-top beam has constant intensity inside radius $a$ and negligible intensity outside:

$$A=\pi a^2,I=\frac{P}{\pi a^2}.$$

Here the intensity is both the local intensity throughout the illuminated disk and the average intensity over that disk.

Gaussian beam

For a fundamental Gaussian beam at its waist,

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

where $w$ is the $1/e^2$ intensity radius. Integrating over the transverse plane gives

$$P=\frac{\pi w^2}{2}{I}_0,$$

so the on-axis, or peak, intensity is

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

This factor of two is important. A Gaussian beam of power $P$ and quoted radius $w$ has twice the on-axis intensity of a uniform disk having area $\pi w^2$.

It is sometimes useful to define an effective area relative to the peak intensity:

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

This is not the physical area bounded by a sharp edge; it is a convenient way to preserve the relationship $P=I_0{A}_{\mathrm{eff}}$.

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From optical energy to photon number

At optical frequency $\nu$, every photon carries

$$E_{\gamma }=h\nu =\frac{hc}{\lambda }.$$

The distinction is central:

• Intensity tells you how much energy crosses unit area per second.
• Photon flux tells you how many photons cross unit area per second.
• Photon rate tells you how many photons cross the entire beam per second.

For monochromatic or sufficiently narrowband light,

$${\Phi }_{\gamma }=\frac{I}{h\nu }=\frac{I\lambda }{hc},$$

where ${\Phi }_{\gamma }$ has units $\mathrm{photons}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$.

The total photon rate is

$$\dot{N}=\frac{P}{h\nu }=\frac{P\lambda }{hc}.$$

Equivalently, for a flat-top beam,

$$\dot{N}=A{\Phi }_{\gamma }.$$

For a Gaussian beam, ${\Phi }_{\gamma }(r)$ follows the Gaussian intensity profile, but the total photon rate still depends only on total power and photon energy.

A useful optical conversion is

$$E_{\gamma }[\mathrm{eV}]\approx \frac{1239.84}{\lambda [\mathrm{nm}]}.$$

Thus, a $1064\mathrm{nm}$ photon has energy about $1.17\mathrm{eV}$, whereas a $532\mathrm{nm}$ photon has twice that energy. At fixed laser power, frequency doubling therefore halves the photon rate.

For a time-dependent beam envelope, these expressions apply instant by instant:

$$\dot{N}(t)=\frac{P(t)}{h\nu }.$$

The total photon number in a narrowband pulse of energy $U$ is simply

$$N=\frac{U}{h\nu }.$$

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Spectral quantities: keeping track of bandwidth

The preceding expressions assumed all photons have essentially the same energy. Real laser light has finite linewidth, and broadband sources may occupy a large wavelength range. Then it is necessary to specify how energy is distributed over frequency.

Define:

$$u_{\nu }=\frac{du}{d\nu },I_{\nu }=\frac{dI}{d\nu },P_{\nu }=\frac{dP}{d\nu }.$$

Their units are:

Quantity	Meaning	SI units
$u_{\nu }$	spectral energy density	$\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$
$I_{\nu }$	spectral intensity	$\mathrm{W}{\mathrm{m}}^{-2}\mathrm{H}{\mathrm{z}}^{-1}$
$P_{\nu }$	power spectral density	$\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1}$

For a directed beam in vacuum,

$$I_{\nu }=c{u}_{\nu }.$$

Integrating over the occupied spectral bandwidth gives the ordinary, frequency-integrated quantities:

$$u=\int u_{\nu }d\nu ,I=\int I_{\nu }d\nu ,P=\int P_{\nu }d\nu .$$

If the beam has a uniform transverse profile of area $A$,

$$P_{\nu }=A{I}_{\nu }.$$

For a sufficiently narrow spectrum centered on ${\nu }_0$, with bandwidth $\Delta \nu$,

$$I\approx I_{\nu }({\nu }_0)\Delta \nu ,u\approx u_{\nu }({\nu }_0)\Delta \nu .$$

The approximation means that the spectral density varies little over the bandwidth.

There is a terminology trap worth flagging. In astrophysical radiative transfer, $I_{\nu }$ often denotes specific intensity, with an additional $\mathrm{s}{\mathrm{r}}^{-1}$ in its units. In this course, $I_{\nu }$ means the spectral density of optical intensity, or irradiance, of a collimated beam unless stated otherwise. Whenever “intensity” appears, inspect its units.

Frequency spectra versus wavelength spectra

Spectra can be expressed per unit frequency or per unit wavelength:

$$u_{\lambda }=\frac{du}{d\lambda },I_{\lambda }=\frac{dI}{d\lambda },P_{\lambda }=\frac{dP}{d\lambda }.$$

Because

$$\nu =\frac{c}{\lambda },$$

the spectral densities are related by a Jacobian:

$$u_{\lambda }=\frac{c}{{\lambda }^2}{u}_{\nu },I_{\lambda }=\frac{c}{{\lambda }^2}{I}_{\nu },P_{\lambda }=\frac{c}{{\lambda }^2}{P}_{\nu }.$$

The factor $c/{\lambda }^2$ is essential. Equal intervals in wavelength do not correspond to equal intervals in frequency. A spectrum that appears symmetric when plotted against wavelength generally does not remain symmetric when replotted against frequency.

The same conversion gives the magnitude of a narrow wavelength bandwidth:

$$\Delta \nu \approx \frac{c}{{\lambda }_0^2}\Delta \lambda .$$

The minus sign from differentiating $\nu =c/\lambda$ only records that frequency decreases as wavelength increases; bandwidths are positive magnitudes.

The plotted spectrum uses logarithmic $dBm/THz$ units. Before integrating such data, convert every plotted value to linear spectral power density. If $L$ is the plotted value in $dBm/THz$,

$$P_{\nu }[\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1}]=\frac{10^{(L-30)/10}}{10^{12}}.$$

You must also use $d\nu$, not $d\lambda$, when integrating $P_{\nu }$. If the data are tabulated against wavelength, write the total power as

$$P=\int P_{\nu }(\lambda )\frac{c}{{\lambda }^2}d\lambda .$$

The apparent area under a logarithmic plot is therefore not itself a physical power.

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A complete conversion example

Consider a continuous-wave $1064\mathrm{nm}$ laser with:

• total power $P=2.0\mathrm{W}$,
• flat-top radius $a=0.50\mathrm{mm}$,
• narrow wavelength width $\Delta \lambda =0.20\mathrm{nm}$.

First calculate beam area:

$$A=\pi a^2=\pi (5.0\times 10^{-4}\mathrm{m}{)}^2=7.85\times 10^{-7}{\mathrm{m}}^2.$$

The optical intensity is

$$I=\frac{P}{A}=2.55\times 10^6\mathrm{W}{\mathrm{m}}^{-2}.$$

Equivalently,

$$I=255\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}.$$

The photon energy is

$$E_{\gamma }=\frac{hc}{\lambda }=1.87\times 10^{-19}\mathrm{J}\approx 1.17\mathrm{eV}.$$

So the photon flux is

$${\Phi }_{\gamma }=\frac{I}{E_{\gamma }}=1.36\times 10^{25}\mathrm{photons}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}.$$

The total photon rate is

$$\dot{N}=\frac{P}{E_{\gamma }}=1.07\times 10^{19}\mathrm{photons}{\mathrm{s}}^{-1}.$$

As a consistency check,

$$A{\Phi }_{\gamma }=7.85\times 10^{-7}\times 1.36\times 10^{25}=1.07\times 10^{19}\mathrm{photons}{\mathrm{s}}^{-1},$$

as required.

Now convert the wavelength width to a frequency bandwidth:

$$\Delta \nu \approx \frac{c}{{\lambda }^2}\Delta \lambda =5.3\times 10^{10}\mathrm{Hz}.$$

This is about $53\mathrm{GHz}$. Because the fractional bandwidth is small, use the narrowband approximation:

$$I_{\nu }\approx \frac{I}{\Delta \nu }=4.8\times 10^{-5}\mathrm{W}{\mathrm{m}}^{-2}\mathrm{H}{\mathrm{z}}^{-1}.$$

Finally, the corresponding free-space spectral energy density is

$$u_{\nu }=\frac{I_{\nu }}{c}=1.6\times 10^{-13}\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}.$$

All quantities describe the same beam:

Description	Result
Total power	$2.0\mathrm{W}$
Intensity	$2.55\times 10^6\mathrm{W}{\mathrm{m}}^{-2}$
Photon energy	$1.87\times 10^{-19}\mathrm{J}$
Photon flux	$1.36\times 10^{25}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$
Photon rate	$1.07\times 10^{19}{\mathrm{s}}^{-1}$
Frequency bandwidth	$5.3\times 10^{10}\mathrm{Hz}$
Spectral energy density in vacuum	$1.6\times 10^{-13}\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$

If the same $2.0\mathrm{W}$ beam were Gaussian with $w=0.50\mathrm{mm}$, its on-axis intensity and photon flux would each be twice the flat-top values, while its total photon rate would remain exactly $1.07\times 10^{19}{\mathrm{s}}^{-1}$.

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A reliable conversion workflow

When faced with a laser-radiation calculation, make the choices in this order:

1. Identify whether the given quantity is total or local.
   Power is total; intensity and photon flux are local. Use the transverse beam profile to connect them.

2. Identify whether the quantity is spectrally integrated.
   A quantity with $\mathrm{H}{\mathrm{z}}^{-1}$, $\mathrm{TH}{\mathrm{z}}^{-1}$, $\mathrm{n}{\mathrm{m}}^{-1}$, or similar units needs integration over bandwidth to obtain a total.

3. Keep frequency and wavelength representations consistent.
   Convert with the $c/{\lambda }^2$ Jacobian when changing spectral coordinates.

4. Convert energy transport to photon transport using $h\nu$.
   Divide power or intensity by photon energy to obtain photon rate or flux.

5. Only then convert between energy density and intensity.
   For a directed vacuum beam, use $I=cu$; in a medium, use $I=v_{g}u$ when the total electromagnetic-plus-material energy density is intended.

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Takeaways and next step

You should now be able to move among four complementary descriptions of a beam:

$$P\text{and}I$$

describe energy transport globally and locally;

$$\dot{N}\text{and}{\Phi }_{\gamma }$$

describe photon transport globally and locally; and

$$u_{\nu },I_{\nu },P_{\nu }$$

retain the spectral resolution needed for finite-linewidth radiation.

The essential identities are

$$I=\frac{P}{A}$$

for a uniform beam,

$${\Phi }_{\gamma }=\frac{I}{h\nu },\dot{N}=\frac{P}{h\nu },$$

and, in vacuum,

$$I_{\nu }=c{u}_{\nu }.$$

Next, we will use this quantitative language to distinguish absorption, spontaneous emission, and stimulated emission—not only by their level transitions, but also by the properties of the radiation they produce or respond to.