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# Optical Intensity

Definition: optical power per unit area

German: optische Intensität

Units: W/m2, W/cm2

Formula symbol: <$I$>

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The term optical intensity (or just intensity) is quite common in optical physics and technology, but it is somewhat problematic, since it is used with substantially different meanings. One may recommend using optical intensity only in a non-quantitative way and only use well-defined radiometric quantities like radiant intensity and irradiance for quantitative references. However, we must deal with literature using the term in different ways, as explained in the following sections. Certainly, for quantitative reference it is important to clearly indicate which meaning of intensity is used.

## Non-quantitative and Inaccurate Meanings of Intensity

The term intensity is often used in a non-quantitative way. For example, such a statement could be “high intensity laser beams are used for laser material processing” – just like “the sun is very bright today”. An actual numerical value is not specified in such cases. Such non-quantitative statements rarely create a risk of misunderstanding.

In other cases, intensities are meant to be quantitative measures, but used in quite inaccurate ways. Sometimes, what is actually meant is for example an optical power, an irradiance or a radiant intensity – these are examples of very different quantities which should not be confused. Such uses of the term should be avoided, but in certain contexts they are nevertheless very common. An example is that the intensity noise of a laser normally refers to noise (fluctuations) of its optical power rather than e.g. an irradiance within its beam profile.

## Intensities in Optical Physics

In optical physics, the intensity <$I$>, e.g. of a laser beam at some location, is generally understood to the optical power per unit area, which is transmitted through an imagined surface perpendicular to the propagation direction. The units of the optical intensity (or light intensity) are W/m2 or (more commonly) W/cm2. The intensity is the product of photon energy and photon flux. It is sometimes called optical energy flux.

For a monochromatic propagating wave, such as a plane wave or a Gaussian beam, the local intensity is related to the amplitude <$E$> of the electric field via

$$I = \frac{{{\upsilon _{\rm{p}}}{\varepsilon _0}{\varepsilon _{\rm{r}}}{\mu _{\rm{r}}}}}{2}{\left| E \right|^2} = \frac{{c{\varepsilon _0}n}}{2}{\left| E \right|^2}$$

where <$v_\textrm{p}$> is the phase velocity, <$c$> is the vacuum velocity of light, and <$n$> is the refractive index. For non-monochromatic waves, the intensity contributions of different spectral components can simply be added, if beat notes are not of interest.

Optical intensities and powers are normally understood as quantities which are averaged over at least one oscillation cycle. In other words, they are not considered to be oscillating on the time scale of an optical oscillation.

Note that the above equation does not hold for arbitrary electromagnetic fields. For example, an evanescent wave may have a finite electrical amplitude while not transferring any power. The intensity should then be defined as the magnitude of the Poynting vector.

When light is received by a surface, an optical intensity causes an irradiance, which is the intensity times the cosine of the angle against normal direction.

In laser technology, one frequently assumes the same meaning of intensity as an optical physics. For a laser beam with a flat-top intensity profile (i.e., with a constant intensity over some area, and zero intensity outside), the intensity is simply the optical power <$P$> divided by the beam area. For a Gaussian beam with optical power <$P$> and Gaussian beam radius <$w$>, the peak intensity (on the beam axis) is

$${I_{\rm{p}}} = \frac{P}{{\pi {w^2}/2}}$$

which is two times higher than is often assumed. The equation can be verified by integrating the intensity over the whole beam area, which must result in the total power.

In a multimode laser beam, generated in a laser where higher-order transverse resonator modes are excited, the shape of the transverse intensity profile can undergo significant changes as the relative optical phases of the modes change with time. The peak intensity can then vary, and may occur at locations at some distance from the beam axis.

Optical intensities with the meaning as used in optical physics are relevant in various situations:

Beam profilers can be used for measuring the shape of the intensity profile of a laser beam.

In radiometry, the radiant intensity is understood to be the radiant flux (or radiant power) per unit solid angle, which leads to units of W/sr (watts per steradian). This meaning is obviously quite different from that of intensities in optical physics. It is usually applied with the approximation of a point source, i.e., for observation distances which are much larger than the spatial extension of the light source, where the wavefronts are approximately spherical.

At a distance <$d$> from a source with radiant intensity <$I$>, an area element with its normal direction towards the source receives an irradiance <$E = I / d^2$>.

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## Questions and Comments from Users

2020-10-10

In many textbooks and articles discussing derivations of the nonlinear Schrödinger equation for pulse propagation, for example in optical fibers, the electric field <$E$> is expressed as <$E = (1/2) A \exp(i k z - i \omega t) + c.c.$>, where <$A$> is the complex slowly varying amplitude. In many of these papers, <$|A|^2$> is then equated directly to the power. However, this appears to be dimensionally incorrect. Shouldn't it be, instead, <$|A|^2 = b P$>, where <$b = 4 / (c * n * \epsilon_0 A_\rm{eff})$>, where <$\epsilon_0$> is the vacuum permittivity and <$A_\rm{eff}$> the effective area?

Well, this is just an example for the use of the term intensity with a vague meaning, where people don't care so much about the absolute scaling. Their “intensity” is e.g. taken to be just the E-field intensity squared, or in other cases it is normalized such that its modulus squared integrated over some cross-section is unity.

2023-11-07

According to the formula given, the intensity of light is completely independent of the frequency. However, this would mean, that two electromagnetic waves of the same amplitude but with different frequencies would have the same intensity, which shouldn't be the case as the higher frequency wave carries more energy. Why is this formula still valid?

For a given electric field amplitude, the intensity indeed does not additionally depend on the optical frequency. I don't know a rule saying that a higher frequency wave carries more energy. Only, for an oscillating electric dipole, for example, you get a higher radiated electric field amplitude (and consequently a higher intensity) for a given amplitude if the frequency is higher. Another aspect is that the quantum energy <$h\nu$> scales with the frequency, but that is just the energy per photon.

2023-12-31

According to the website linked on the word “irradiance”, irradiance is in units W/m2. However, in the formula given in the last sentence, it states that irradiance is given by radiant intensity divided by distance squared. It also says that radiant intensity is in units W/sr. This would result in irradiance being in units W/sr/m2, which are the units of radiance and not of irradiance. Shouldn't the formula show irradiance be equal to radiant flux divided by distance squared instead?

No, the article text is correct. Note that radians stand for a ratio, and are thus not real units. That resolves your problem with the units. With your suggested formula, you would be off by a factor of <$4\pi$>, as in a distance <$d$> from the source the radiant flux goes through a spherical surface of magnitude <$4\pi \; d^2$>.

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